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Theorem List for Metamath Proof Explorer - 20101-20200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsrgcom 20101 Commutativity of the additive group of a semiring. (Contributed by Thierry Arnoux, 1-Apr-2018.)
𝐡 = (Baseβ€˜π‘…)    &    + = (+gβ€˜π‘…)    β‡’   ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 + π‘Œ) = (π‘Œ + 𝑋))
 
Theoremsrgrz 20102 The zero of a semiring is a right-absorbing element. (Contributed by Thierry Arnoux, 1-Apr-2018.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    β‡’   ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐡) β†’ (𝑋 Β· 0 ) = 0 )
 
Theoremsrglz 20103 The zero of a semiring is a left-absorbing element. (Contributed by AV, 23-Aug-2019.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    β‡’   ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐡) β†’ ( 0 Β· 𝑋) = 0 )
 
Theoremsrgisid 20104* In a semiring, the only left-absorbing element is the additive identity. Remark in [Golan] p. 1. (Contributed by Thierry Arnoux, 1-May-2018.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ SRing)    &   (πœ‘ β†’ 𝑍 ∈ 𝐡)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐡) β†’ (𝑍 Β· π‘₯) = 𝑍)    β‡’   (πœ‘ β†’ 𝑍 = 0 )
 
Theoremo2timesd 20105* An element of a ring-like structure plus itself is two times the element. "Two" in such a structure is the sum of the unity element with itself. This (formerly) part of the proof for ringcom 20169 depends on the (right) distributivity and the existence of a (left) multiplicative identity only. (Contributed by GΓ©rard Lang, 4-Dec-2014.) (Revised by AV, 1-Feb-2025.)
(πœ‘ β†’ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 βˆ€π‘§ ∈ 𝐡 ((π‘₯ + 𝑦) Β· 𝑧) = ((π‘₯ Β· 𝑧) + (𝑦 Β· 𝑧)))    &   (πœ‘ β†’ 1 ∈ 𝐡)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ 𝐡 ( 1 Β· π‘₯) = π‘₯)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    β‡’   (πœ‘ β†’ (𝑋 + 𝑋) = (( 1 + 1 ) Β· 𝑋))
 
Theoremrglcom4d 20106* Restricted commutativity of the addition in a ring-like structure. This (formerly) part of the proof for ringcom 20169 depends on the closure of the addition, the (left and right) distributivity and the existence of a (left) multiplicative identity only. (Contributed by GΓ©rard Lang, 4-Dec-2014.) (Revised by AV, 1-Feb-2025.)
(πœ‘ β†’ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 βˆ€π‘§ ∈ 𝐡 ((π‘₯ + 𝑦) Β· 𝑧) = ((π‘₯ Β· 𝑧) + (𝑦 Β· 𝑧)))    &   (πœ‘ β†’ 1 ∈ 𝐡)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ 𝐡 ( 1 Β· π‘₯) = π‘₯)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (π‘₯ + 𝑦) ∈ 𝐡)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 βˆ€π‘§ ∈ 𝐡 (π‘₯ Β· (𝑦 + 𝑧)) = ((π‘₯ Β· 𝑦) + (π‘₯ Β· 𝑧)))    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    β‡’   (πœ‘ β†’ ((𝑋 + 𝑋) + (π‘Œ + π‘Œ)) = ((𝑋 + π‘Œ) + (𝑋 + π‘Œ)))
 
Theoremsrgo2times 20107 A semiring element plus itself is two times the element. "Two" in an arbitrary (unital) semiring is the sum of the unity element with itself. (Contributed by AV, 24-Aug-2021.) Variant of o2timesd 20105 for semirings. (Revised by AV, 1-Feb-2025.)
𝐡 = (Baseβ€˜π‘…)    &    + = (+gβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    β‡’   ((𝑅 ∈ SRing ∧ 𝐴 ∈ 𝐡) β†’ (𝐴 + 𝐴) = (( 1 + 1 ) Β· 𝐴))
 
Theoremsrgcom4lem 20108 Lemma for srgcom4 20109. This (formerly) part of the proof for ringcom 20169 is applicable for semirings (without using the commutativity of the addition given per definition of a semiring). (Contributed by GΓ©rard Lang, 4-Dec-2014.) (Revised by AV, 1-Feb-2025.) (Proof modification is discouraged.)
𝐡 = (Baseβ€˜π‘…)    &    + = (+gβ€˜π‘…)    β‡’   ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((𝑋 + 𝑋) + (π‘Œ + π‘Œ)) = ((𝑋 + π‘Œ) + (𝑋 + π‘Œ)))
 
Theoremsrgcom4 20109 Restricted commutativity of the addition in semirings (without using the commutativity of the addition given per definition of a semiring). (Contributed by AV, 1-Feb-2025.) (Proof modification is discouraged.)
𝐡 = (Baseβ€˜π‘…)    &    + = (+gβ€˜π‘…)    β‡’   ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((𝑋 + (𝑋 + π‘Œ)) + π‘Œ) = ((𝑋 + (π‘Œ + 𝑋)) + π‘Œ))
 
Theoremsrg1zr 20110 The only semiring with a base set consisting of one element is the zero ring (at least if its operations are internal binary operations). (Contributed by FL, 13-Feb-2010.) (Revised by AV, 25-Jan-2020.)
𝐡 = (Baseβ€˜π‘…)    &    + = (+gβ€˜π‘…)    &    βˆ— = (.rβ€˜π‘…)    β‡’   (((𝑅 ∈ SRing ∧ + Fn (𝐡 Γ— 𝐡) ∧ βˆ— Fn (𝐡 Γ— 𝐡)) ∧ 𝑍 ∈ 𝐡) β†’ (𝐡 = {𝑍} ↔ ( + = {βŸ¨βŸ¨π‘, π‘βŸ©, π‘βŸ©} ∧ βˆ— = {βŸ¨βŸ¨π‘, π‘βŸ©, π‘βŸ©})))
 
Theoremsrgen1zr 20111 The only semiring with one element is the zero ring (at least if its operations are internal binary operations). (Contributed by FL, 14-Feb-2010.) (Revised by AV, 25-Jan-2020.)
𝐡 = (Baseβ€˜π‘…)    &    + = (+gβ€˜π‘…)    &    βˆ— = (.rβ€˜π‘…)    &   π‘ = (0gβ€˜π‘…)    β‡’   ((𝑅 ∈ SRing ∧ + Fn (𝐡 Γ— 𝐡) ∧ βˆ— Fn (𝐡 Γ— 𝐡)) β†’ (𝐡 β‰ˆ 1o ↔ ( + = {βŸ¨βŸ¨π‘, π‘βŸ©, π‘βŸ©} ∧ βˆ— = {βŸ¨βŸ¨π‘, π‘βŸ©, π‘βŸ©})))
 
Theoremsrgmulgass 20112 An associative property between group multiple and ring multiplication for semirings. (Contributed by AV, 23-Aug-2019.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.gβ€˜π‘…)    &    Γ— = (.rβ€˜π‘…)    β‡’   ((𝑅 ∈ SRing ∧ (𝑁 ∈ β„•0 ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ ((𝑁 Β· 𝑋) Γ— π‘Œ) = (𝑁 Β· (𝑋 Γ— π‘Œ)))
 
Theoremsrgpcomp 20113 If two elements of a semiring commute, they also commute if one of the elements is raised to a higher power. (Contributed by AV, 23-Aug-2019.)
𝑆 = (Baseβ€˜π‘…)    &    Γ— = (.rβ€˜π‘…)    &   πΊ = (mulGrpβ€˜π‘…)    &    ↑ = (.gβ€˜πΊ)    &   (πœ‘ β†’ 𝑅 ∈ SRing)    &   (πœ‘ β†’ 𝐴 ∈ 𝑆)    &   (πœ‘ β†’ 𝐡 ∈ 𝑆)    &   (πœ‘ β†’ 𝐾 ∈ β„•0)    &   (πœ‘ β†’ (𝐴 Γ— 𝐡) = (𝐡 Γ— 𝐴))    β‡’   (πœ‘ β†’ ((𝐾 ↑ 𝐡) Γ— 𝐴) = (𝐴 Γ— (𝐾 ↑ 𝐡)))
 
Theoremsrgpcompp 20114 If two elements of a semiring commute, they also commute if the elements are raised to a higher power. (Contributed by AV, 23-Aug-2019.)
𝑆 = (Baseβ€˜π‘…)    &    Γ— = (.rβ€˜π‘…)    &   πΊ = (mulGrpβ€˜π‘…)    &    ↑ = (.gβ€˜πΊ)    &   (πœ‘ β†’ 𝑅 ∈ SRing)    &   (πœ‘ β†’ 𝐴 ∈ 𝑆)    &   (πœ‘ β†’ 𝐡 ∈ 𝑆)    &   (πœ‘ β†’ 𝐾 ∈ β„•0)    &   (πœ‘ β†’ (𝐴 Γ— 𝐡) = (𝐡 Γ— 𝐴))    &   (πœ‘ β†’ 𝑁 ∈ β„•0)    β‡’   (πœ‘ β†’ (((𝑁 ↑ 𝐴) Γ— (𝐾 ↑ 𝐡)) Γ— 𝐴) = (((𝑁 + 1) ↑ 𝐴) Γ— (𝐾 ↑ 𝐡)))
 
Theoremsrgpcomppsc 20115 If two elements of a semiring commute, they also commute if the elements are raised to a higher power and a scalar multiplication is involved. (Contributed by AV, 23-Aug-2019.)
𝑆 = (Baseβ€˜π‘…)    &    Γ— = (.rβ€˜π‘…)    &   πΊ = (mulGrpβ€˜π‘…)    &    ↑ = (.gβ€˜πΊ)    &   (πœ‘ β†’ 𝑅 ∈ SRing)    &   (πœ‘ β†’ 𝐴 ∈ 𝑆)    &   (πœ‘ β†’ 𝐡 ∈ 𝑆)    &   (πœ‘ β†’ 𝐾 ∈ β„•0)    &   (πœ‘ β†’ (𝐴 Γ— 𝐡) = (𝐡 Γ— 𝐴))    &   (πœ‘ β†’ 𝑁 ∈ β„•0)    &    Β· = (.gβ€˜π‘…)    &   (πœ‘ β†’ 𝐢 ∈ β„•0)    β‡’   (πœ‘ β†’ ((𝐢 Β· ((𝑁 ↑ 𝐴) Γ— (𝐾 ↑ 𝐡))) Γ— 𝐴) = (𝐢 Β· (((𝑁 + 1) ↑ 𝐴) Γ— (𝐾 ↑ 𝐡))))
 
Theoremsrglmhm 20116* Left-multiplication in a semiring by a fixed element of the ring is a monoid homomorphism, analogous to ringlghm 20201. (Contributed by AV, 23-Aug-2019.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    β‡’   ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐡) β†’ (π‘₯ ∈ 𝐡 ↦ (𝑋 Β· π‘₯)) ∈ (𝑅 MndHom 𝑅))
 
Theoremsrgrmhm 20117* Right-multiplication in a semiring by a fixed element of the ring is a monoid homomorphism, analogous to ringrghm 20202. (Contributed by AV, 23-Aug-2019.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    β‡’   ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐡) β†’ (π‘₯ ∈ 𝐡 ↦ (π‘₯ Β· 𝑋)) ∈ (𝑅 MndHom 𝑅))
 
Theoremsrgsummulcr 20118* A finite semiring sum multiplied by a constant, analogous to gsummulc1 20205. (Contributed by AV, 23-Aug-2019.)
𝐡 = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &    + = (+gβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ SRing)    &   (πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ (π‘˜ ∈ 𝐴 ↦ 𝑋) finSupp 0 )    β‡’   (πœ‘ β†’ (𝑅 Ξ£g (π‘˜ ∈ 𝐴 ↦ (𝑋 Β· π‘Œ))) = ((𝑅 Ξ£g (π‘˜ ∈ 𝐴 ↦ 𝑋)) Β· π‘Œ))
 
Theoremsgsummulcl 20119* A finite semiring sum multiplied by a constant, analogous to gsummulc2 20206. (Contributed by AV, 23-Aug-2019.)
𝐡 = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &    + = (+gβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ SRing)    &   (πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ (π‘˜ ∈ 𝐴 ↦ 𝑋) finSupp 0 )    β‡’   (πœ‘ β†’ (𝑅 Ξ£g (π‘˜ ∈ 𝐴 ↦ (π‘Œ Β· 𝑋))) = (π‘Œ Β· (𝑅 Ξ£g (π‘˜ ∈ 𝐴 ↦ 𝑋))))
 
Theoremsrg1expzeq1 20120 The exponentiation (by a nonnegative integer) of the multiplicative identity of a semiring, analogous to mulgnn0z 19018. (Contributed by AV, 25-Nov-2019.)
𝐺 = (mulGrpβ€˜π‘…)    &    Β· = (.gβ€˜πΊ)    &    1 = (1rβ€˜π‘…)    β‡’   ((𝑅 ∈ SRing ∧ 𝑁 ∈ β„•0) β†’ (𝑁 Β· 1 ) = 1 )
 
10.3.4.1  The binomial theorem for semirings

In this section, we prove the binomial theorem for semirings, srgbinom 20126, which is a generalization of the binomial theorem for complex numbers, binom 15781: (𝐴 + 𝐡)↑𝑁 is the sum from π‘˜ = 0 to 𝑁 of (𝑁Cπ‘˜) Β· ((π΄β†‘π‘˜) Β· (𝐡↑(𝑁 βˆ’ π‘˜)).

Note that the binomial theorem also holds in the non-unital case (that is, in a "rg") and actually, the additive identity is not needed in its proof either. Therefore, it can be proven in even more general cases. An example is the "rg" (resp. "rg without a zero") of integrable nonnegative (resp. positive) functions on ℝ.

Special cases of the binomial theorem are csrgbinom 20127 (binomial theorem for commutative semirings) and crngbinom 20224 (binomial theorem for commutative rings).

 
Theoremsrgbinomlem1 20121 Lemma 1 for srgbinomlem 20125. (Contributed by AV, 23-Aug-2019.)
𝑆 = (Baseβ€˜π‘…)    &    Γ— = (.rβ€˜π‘…)    &    Β· = (.gβ€˜π‘…)    &    + = (+gβ€˜π‘…)    &   πΊ = (mulGrpβ€˜π‘…)    &    ↑ = (.gβ€˜πΊ)    &   (πœ‘ β†’ 𝑅 ∈ SRing)    &   (πœ‘ β†’ 𝐴 ∈ 𝑆)    &   (πœ‘ β†’ 𝐡 ∈ 𝑆)    &   (πœ‘ β†’ (𝐴 Γ— 𝐡) = (𝐡 Γ— 𝐴))    &   (πœ‘ β†’ 𝑁 ∈ β„•0)    β‡’   ((πœ‘ ∧ (𝐷 ∈ β„•0 ∧ 𝐸 ∈ β„•0)) β†’ ((𝐷 ↑ 𝐴) Γ— (𝐸 ↑ 𝐡)) ∈ 𝑆)
 
Theoremsrgbinomlem2 20122 Lemma 2 for srgbinomlem 20125. (Contributed by AV, 23-Aug-2019.)
𝑆 = (Baseβ€˜π‘…)    &    Γ— = (.rβ€˜π‘…)    &    Β· = (.gβ€˜π‘…)    &    + = (+gβ€˜π‘…)    &   πΊ = (mulGrpβ€˜π‘…)    &    ↑ = (.gβ€˜πΊ)    &   (πœ‘ β†’ 𝑅 ∈ SRing)    &   (πœ‘ β†’ 𝐴 ∈ 𝑆)    &   (πœ‘ β†’ 𝐡 ∈ 𝑆)    &   (πœ‘ β†’ (𝐴 Γ— 𝐡) = (𝐡 Γ— 𝐴))    &   (πœ‘ β†’ 𝑁 ∈ β„•0)    β‡’   ((πœ‘ ∧ (𝐢 ∈ β„•0 ∧ 𝐷 ∈ β„•0 ∧ 𝐸 ∈ β„•0)) β†’ (𝐢 Β· ((𝐷 ↑ 𝐴) Γ— (𝐸 ↑ 𝐡))) ∈ 𝑆)
 
Theoremsrgbinomlem3 20123* Lemma 3 for srgbinomlem 20125. (Contributed by AV, 23-Aug-2019.) (Proof shortened by AV, 27-Oct-2019.)
𝑆 = (Baseβ€˜π‘…)    &    Γ— = (.rβ€˜π‘…)    &    Β· = (.gβ€˜π‘…)    &    + = (+gβ€˜π‘…)    &   πΊ = (mulGrpβ€˜π‘…)    &    ↑ = (.gβ€˜πΊ)    &   (πœ‘ β†’ 𝑅 ∈ SRing)    &   (πœ‘ β†’ 𝐴 ∈ 𝑆)    &   (πœ‘ β†’ 𝐡 ∈ 𝑆)    &   (πœ‘ β†’ (𝐴 Γ— 𝐡) = (𝐡 Γ— 𝐴))    &   (πœ‘ β†’ 𝑁 ∈ β„•0)    &   (πœ“ β†’ (𝑁 ↑ (𝐴 + 𝐡)) = (𝑅 Ξ£g (π‘˜ ∈ (0...𝑁) ↦ ((𝑁Cπ‘˜) Β· (((𝑁 βˆ’ π‘˜) ↑ 𝐴) Γ— (π‘˜ ↑ 𝐡))))))    β‡’   ((πœ‘ ∧ πœ“) β†’ ((𝑁 ↑ (𝐴 + 𝐡)) Γ— 𝐴) = (𝑅 Ξ£g (π‘˜ ∈ (0...(𝑁 + 1)) ↦ ((𝑁Cπ‘˜) Β· ((((𝑁 + 1) βˆ’ π‘˜) ↑ 𝐴) Γ— (π‘˜ ↑ 𝐡))))))
 
Theoremsrgbinomlem4 20124* Lemma 4 for srgbinomlem 20125. (Contributed by AV, 24-Aug-2019.) (Proof shortened by AV, 19-Nov-2019.)
𝑆 = (Baseβ€˜π‘…)    &    Γ— = (.rβ€˜π‘…)    &    Β· = (.gβ€˜π‘…)    &    + = (+gβ€˜π‘…)    &   πΊ = (mulGrpβ€˜π‘…)    &    ↑ = (.gβ€˜πΊ)    &   (πœ‘ β†’ 𝑅 ∈ SRing)    &   (πœ‘ β†’ 𝐴 ∈ 𝑆)    &   (πœ‘ β†’ 𝐡 ∈ 𝑆)    &   (πœ‘ β†’ (𝐴 Γ— 𝐡) = (𝐡 Γ— 𝐴))    &   (πœ‘ β†’ 𝑁 ∈ β„•0)    &   (πœ“ β†’ (𝑁 ↑ (𝐴 + 𝐡)) = (𝑅 Ξ£g (π‘˜ ∈ (0...𝑁) ↦ ((𝑁Cπ‘˜) Β· (((𝑁 βˆ’ π‘˜) ↑ 𝐴) Γ— (π‘˜ ↑ 𝐡))))))    β‡’   ((πœ‘ ∧ πœ“) β†’ ((𝑁 ↑ (𝐴 + 𝐡)) Γ— 𝐡) = (𝑅 Ξ£g (π‘˜ ∈ (0...(𝑁 + 1)) ↦ ((𝑁C(π‘˜ βˆ’ 1)) Β· ((((𝑁 + 1) βˆ’ π‘˜) ↑ 𝐴) Γ— (π‘˜ ↑ 𝐡))))))
 
Theoremsrgbinomlem 20125* Lemma for srgbinom 20126. Inductive step, analogous to binomlem 15780. (Contributed by AV, 24-Aug-2019.)
𝑆 = (Baseβ€˜π‘…)    &    Γ— = (.rβ€˜π‘…)    &    Β· = (.gβ€˜π‘…)    &    + = (+gβ€˜π‘…)    &   πΊ = (mulGrpβ€˜π‘…)    &    ↑ = (.gβ€˜πΊ)    &   (πœ‘ β†’ 𝑅 ∈ SRing)    &   (πœ‘ β†’ 𝐴 ∈ 𝑆)    &   (πœ‘ β†’ 𝐡 ∈ 𝑆)    &   (πœ‘ β†’ (𝐴 Γ— 𝐡) = (𝐡 Γ— 𝐴))    &   (πœ‘ β†’ 𝑁 ∈ β„•0)    &   (πœ“ β†’ (𝑁 ↑ (𝐴 + 𝐡)) = (𝑅 Ξ£g (π‘˜ ∈ (0...𝑁) ↦ ((𝑁Cπ‘˜) Β· (((𝑁 βˆ’ π‘˜) ↑ 𝐴) Γ— (π‘˜ ↑ 𝐡))))))    β‡’   ((πœ‘ ∧ πœ“) β†’ ((𝑁 + 1) ↑ (𝐴 + 𝐡)) = (𝑅 Ξ£g (π‘˜ ∈ (0...(𝑁 + 1)) ↦ (((𝑁 + 1)Cπ‘˜) Β· ((((𝑁 + 1) βˆ’ π‘˜) ↑ 𝐴) Γ— (π‘˜ ↑ 𝐡))))))
 
Theoremsrgbinom 20126* The binomial theorem for commuting elements of a semiring: (𝐴 + 𝐡)↑𝑁 is the sum from π‘˜ = 0 to 𝑁 of (𝑁Cπ‘˜) Β· ((π΄β†‘π‘˜) Β· (𝐡↑(𝑁 βˆ’ π‘˜)) (generalization of binom 15781). (Contributed by AV, 24-Aug-2019.)
𝑆 = (Baseβ€˜π‘…)    &    Γ— = (.rβ€˜π‘…)    &    Β· = (.gβ€˜π‘…)    &    + = (+gβ€˜π‘…)    &   πΊ = (mulGrpβ€˜π‘…)    &    ↑ = (.gβ€˜πΊ)    β‡’   (((𝑅 ∈ SRing ∧ 𝑁 ∈ β„•0) ∧ (𝐴 ∈ 𝑆 ∧ 𝐡 ∈ 𝑆 ∧ (𝐴 Γ— 𝐡) = (𝐡 Γ— 𝐴))) β†’ (𝑁 ↑ (𝐴 + 𝐡)) = (𝑅 Ξ£g (π‘˜ ∈ (0...𝑁) ↦ ((𝑁Cπ‘˜) Β· (((𝑁 βˆ’ π‘˜) ↑ 𝐴) Γ— (π‘˜ ↑ 𝐡))))))
 
Theoremcsrgbinom 20127* The binomial theorem for commutative semirings. (Contributed by AV, 24-Aug-2019.)
𝑆 = (Baseβ€˜π‘…)    &    Γ— = (.rβ€˜π‘…)    &    Β· = (.gβ€˜π‘…)    &    + = (+gβ€˜π‘…)    &   πΊ = (mulGrpβ€˜π‘…)    &    ↑ = (.gβ€˜πΊ)    β‡’   (((𝑅 ∈ SRing ∧ 𝐺 ∈ CMnd ∧ 𝑁 ∈ β„•0) ∧ (𝐴 ∈ 𝑆 ∧ 𝐡 ∈ 𝑆)) β†’ (𝑁 ↑ (𝐴 + 𝐡)) = (𝑅 Ξ£g (π‘˜ ∈ (0...𝑁) ↦ ((𝑁Cπ‘˜) Β· (((𝑁 βˆ’ π‘˜) ↑ 𝐴) Γ— (π‘˜ ↑ 𝐡))))))
 
10.3.5  Unital rings
 
Syntaxcrg 20128 Extend class notation with class of all (unital) rings.
class Ring
 
Syntaxccrg 20129 Extend class notation with class of all (unital) commutative rings.
class CRing
 
Definitiondf-ring 20130* Define class of all (unital) rings. A unital ring is a set equipped with two everywhere-defined internal operations, whose first one is an additive group structure and the second one is a multiplicative monoid structure, and where the addition is left- and right-distributive for the multiplication. Definition 1 in [BourbakiAlg1] p. 92 or definition of a ring with identity in part Preliminaries of [Roman] p. 19. So that the additive structure must be abelian (see ringcom 20169), care must be taken that in the case of a non-unital ring, the commutativity of addition must be postulated and cannot be proved from the other conditions. Therefore, it can be shown that a unital ring is a non-unital ring (ringrng 20174) only after ringabl 20170 was proven. (Contributed by NM, 18-Oct-2012.) (Revised by Mario Carneiro, 27-Dec-2014.)
Ring = {𝑓 ∈ Grp ∣ ((mulGrpβ€˜π‘“) ∈ Mnd ∧ [(Baseβ€˜π‘“) / π‘Ÿ][(+gβ€˜π‘“) / 𝑝][(.rβ€˜π‘“) / 𝑑]βˆ€π‘₯ ∈ π‘Ÿ βˆ€π‘¦ ∈ π‘Ÿ βˆ€π‘§ ∈ π‘Ÿ ((π‘₯𝑑(𝑦𝑝𝑧)) = ((π‘₯𝑑𝑦)𝑝(π‘₯𝑑𝑧)) ∧ ((π‘₯𝑝𝑦)𝑑𝑧) = ((π‘₯𝑑𝑧)𝑝(𝑦𝑑𝑧))))}
 
Definitiondf-cring 20131 Define class of all commutative rings. (Contributed by Mario Carneiro, 7-Jan-2015.)
CRing = {𝑓 ∈ Ring ∣ (mulGrpβ€˜π‘“) ∈ CMnd}
 
Theoremisring 20132* The predicate "is a (unital) ring". Definition of "ring with unit" in [Schechter] p. 187. (Contributed by NM, 18-Oct-2012.) (Revised by Mario Carneiro, 6-Jan-2015.)
𝐡 = (Baseβ€˜π‘…)    &   πΊ = (mulGrpβ€˜π‘…)    &    + = (+gβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    β‡’   (𝑅 ∈ Ring ↔ (𝑅 ∈ Grp ∧ 𝐺 ∈ Mnd ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 βˆ€π‘§ ∈ 𝐡 ((π‘₯ Β· (𝑦 + 𝑧)) = ((π‘₯ Β· 𝑦) + (π‘₯ Β· 𝑧)) ∧ ((π‘₯ + 𝑦) Β· 𝑧) = ((π‘₯ Β· 𝑧) + (𝑦 Β· 𝑧)))))
 
Theoremringgrp 20133 A ring is a group. (Contributed by NM, 15-Sep-2011.)
(𝑅 ∈ Ring β†’ 𝑅 ∈ Grp)
 
Theoremringmgp 20134 A ring is a monoid under multiplication. (Contributed by Mario Carneiro, 6-Jan-2015.)
𝐺 = (mulGrpβ€˜π‘…)    β‡’   (𝑅 ∈ Ring β†’ 𝐺 ∈ Mnd)
 
Theoremiscrng 20135 A commutative ring is a ring whose multiplication is a commutative monoid. (Contributed by Mario Carneiro, 7-Jan-2015.)
𝐺 = (mulGrpβ€˜π‘…)    β‡’   (𝑅 ∈ CRing ↔ (𝑅 ∈ Ring ∧ 𝐺 ∈ CMnd))
 
Theoremcrngmgp 20136 A commutative ring's multiplication operation is commutative. (Contributed by Mario Carneiro, 7-Jan-2015.)
𝐺 = (mulGrpβ€˜π‘…)    β‡’   (𝑅 ∈ CRing β†’ 𝐺 ∈ CMnd)
 
Theoremringgrpd 20137 A ring is a group. (Contributed by SN, 16-May-2024.)
(πœ‘ β†’ 𝑅 ∈ Ring)    β‡’   (πœ‘ β†’ 𝑅 ∈ Grp)
 
Theoremringmnd 20138 A ring is a monoid under addition. (Contributed by Mario Carneiro, 7-Jan-2015.)
(𝑅 ∈ Ring β†’ 𝑅 ∈ Mnd)
 
Theoremringmgm 20139 A ring is a magma. (Contributed by AV, 31-Jan-2020.)
(𝑅 ∈ Ring β†’ 𝑅 ∈ Mgm)
 
Theoremcrngring 20140 A commutative ring is a ring. (Contributed by Mario Carneiro, 7-Jan-2015.)
(𝑅 ∈ CRing β†’ 𝑅 ∈ Ring)
 
Theoremcrngringd 20141 A commutative ring is a ring. (Contributed by SN, 16-May-2024.)
(πœ‘ β†’ 𝑅 ∈ CRing)    β‡’   (πœ‘ β†’ 𝑅 ∈ Ring)
 
Theoremcrnggrpd 20142 A commutative ring is a group. (Contributed by SN, 16-May-2024.)
(πœ‘ β†’ 𝑅 ∈ CRing)    β‡’   (πœ‘ β†’ 𝑅 ∈ Grp)
 
Theoremmgpf 20143 Restricted functionality of the multiplicative group on rings. (Contributed by Mario Carneiro, 11-Mar-2015.)
(mulGrp β†Ύ Ring):Ring⟢Mnd
 
Theoremringdilem 20144 Properties of a unital ring. (Contributed by NM, 26-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
𝐡 = (Baseβ€˜π‘…)    &    + = (+gβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ ((𝑋 Β· (π‘Œ + 𝑍)) = ((𝑋 Β· π‘Œ) + (𝑋 Β· 𝑍)) ∧ ((𝑋 + π‘Œ) Β· 𝑍) = ((𝑋 Β· 𝑍) + (π‘Œ Β· 𝑍))))
 
Theoremringcl 20145 Closure of the multiplication operation of a ring. (Contributed by NM, 26-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 Β· π‘Œ) ∈ 𝐡)
 
Theoremcrngcom 20146 A commutative ring's multiplication operation is commutative. (Contributed by Mario Carneiro, 7-Jan-2015.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    β‡’   ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 Β· π‘Œ) = (π‘Œ Β· 𝑋))
 
Theoremiscrng2 20147* A commutative ring is a ring whose multiplication is a commutative monoid. (Contributed by Mario Carneiro, 15-Jun-2015.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    β‡’   (𝑅 ∈ CRing ↔ (𝑅 ∈ Ring ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (π‘₯ Β· 𝑦) = (𝑦 Β· π‘₯)))
 
Theoremringass 20148 Associative law for multiplication in a ring. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ ((𝑋 Β· π‘Œ) Β· 𝑍) = (𝑋 Β· (π‘Œ Β· 𝑍)))
 
Theoremringideu 20149* The unity element of a ring is unique. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    β‡’   (𝑅 ∈ Ring β†’ βˆƒ!𝑒 ∈ 𝐡 βˆ€π‘₯ ∈ 𝐡 ((𝑒 Β· π‘₯) = π‘₯ ∧ (π‘₯ Β· 𝑒) = π‘₯))
 
Theoremcrngbascntr 20150 The base set of a commutative ring is its center. (Contributed by SN, 21-Mar-2025.)
𝐡 = (Baseβ€˜πΊ)    &   π‘ = (Cntrβ€˜(mulGrpβ€˜πΊ))    β‡’   (𝐺 ∈ CRing β†’ 𝐡 = 𝑍)
 
Theoremringassd 20151 Associative law for multiplication in a ring. (Contributed by SN, 14-Aug-2024.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    &   (πœ‘ β†’ 𝑍 ∈ 𝐡)    β‡’   (πœ‘ β†’ ((𝑋 Β· π‘Œ) Β· 𝑍) = (𝑋 Β· (π‘Œ Β· 𝑍)))
 
Theoremringcld 20152 Closure of the multiplication operation of a ring. (Contributed by SN, 29-Jul-2024.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    β‡’   (πœ‘ β†’ (𝑋 Β· π‘Œ) ∈ 𝐡)
 
Theoremringdi 20153 Distributive law for the multiplication operation of a ring (left-distributivity). (Contributed by Steve Rodriguez, 9-Sep-2007.)
𝐡 = (Baseβ€˜π‘…)    &    + = (+gβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ (𝑋 Β· (π‘Œ + 𝑍)) = ((𝑋 Β· π‘Œ) + (𝑋 Β· 𝑍)))
 
Theoremringdir 20154 Distributive law for the multiplication operation of a ring (right-distributivity). (Contributed by Steve Rodriguez, 9-Sep-2007.)
𝐡 = (Baseβ€˜π‘…)    &    + = (+gβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ ((𝑋 + π‘Œ) Β· 𝑍) = ((𝑋 Β· 𝑍) + (π‘Œ Β· 𝑍)))
 
Theoremringidcl 20155 The unity element of a ring belongs to the base set of the ring. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)
𝐡 = (Baseβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    β‡’   (𝑅 ∈ Ring β†’ 1 ∈ 𝐡)
 
Theoremring0cl 20156 The zero element of a ring belongs to its base set. (Contributed by Mario Carneiro, 12-Jan-2014.)
𝐡 = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    β‡’   (𝑅 ∈ Ring β†’ 0 ∈ 𝐡)
 
Theoremringidmlem 20157 Lemma for ringlidm 20158 and ringridm 20159. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐡) β†’ (( 1 Β· 𝑋) = 𝑋 ∧ (𝑋 Β· 1 ) = 𝑋))
 
Theoremringlidm 20158 The unity element of a ring is a left multiplicative identity. (Contributed by NM, 15-Sep-2011.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐡) β†’ ( 1 Β· 𝑋) = 𝑋)
 
Theoremringridm 20159 The unity element of a ring is a right multiplicative identity. (Contributed by NM, 15-Sep-2011.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐡) β†’ (𝑋 Β· 1 ) = 𝑋)
 
Theoremisringid 20160* Properties showing that an element 𝐼 is the unity element of a ring. (Contributed by NM, 7-Aug-2013.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    β‡’   (𝑅 ∈ Ring β†’ ((𝐼 ∈ 𝐡 ∧ βˆ€π‘₯ ∈ 𝐡 ((𝐼 Β· π‘₯) = π‘₯ ∧ (π‘₯ Β· 𝐼) = π‘₯)) ↔ 1 = 𝐼))
 
Theoremringlidmd 20161 The unity element of a ring is a left multiplicative identity. (Contributed by SN, 14-Aug-2024.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    β‡’   (πœ‘ β†’ ( 1 Β· 𝑋) = 𝑋)
 
Theoremringridmd 20162 The unity element of a ring is a right multiplicative identity. (Contributed by SN, 14-Aug-2024.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    β‡’   (πœ‘ β†’ (𝑋 Β· 1 ) = 𝑋)
 
Theoremringid 20163* The multiplication operation of a unital ring has (one or more) identity elements. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Mario Carneiro, 22-Dec-2013.) (Revised by AV, 24-Aug-2021.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐡) β†’ βˆƒπ‘’ ∈ 𝐡 ((𝑒 Β· 𝑋) = 𝑋 ∧ (𝑋 Β· 𝑒) = 𝑋))
 
Theoremringo2times 20164 A ring element plus itself is two times the element. "Two" in an arbitrary unital ring is the sum of the unity element with itself. (Contributed by AV, 24-Aug-2021.) Variant of o2timesd 20105 for rings. (Revised by AV, 5-Feb-2025.)
𝐡 = (Baseβ€˜π‘…)    &    + = (+gβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝐴 ∈ 𝐡) β†’ (𝐴 + 𝐴) = (( 1 + 1 ) Β· 𝐴))
 
Theoremringadd2 20165* A ring element plus itself is two times the element. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Mario Carneiro, 22-Dec-2013.) (Revised by AV, 24-Aug-2021.) (Proof shortened by AV, 1-Feb-2025.)
𝐡 = (Baseβ€˜π‘…)    &    + = (+gβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐡) β†’ βˆƒπ‘₯ ∈ 𝐡 (𝑋 + 𝑋) = ((π‘₯ + π‘₯) Β· 𝑋))
 
Theoremringidss 20166 A subset of the multiplicative group has the multiplicative identity as its identity if the identity is in the subset. (Contributed by Mario Carneiro, 27-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
𝑀 = ((mulGrpβ€˜π‘…) β†Ύs 𝐴)    &   π΅ = (Baseβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝐴 βŠ† 𝐡 ∧ 1 ∈ 𝐴) β†’ 1 = (0gβ€˜π‘€))
 
Theoremringacl 20167 Closure of the addition operation of a ring. (Contributed by Mario Carneiro, 14-Jan-2014.)
𝐡 = (Baseβ€˜π‘…)    &    + = (+gβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 + π‘Œ) ∈ 𝐡)
 
Theoremringcomlem 20168 Lemma for ringcom 20169. This (formerly) part of the proof for ringcom 20169 is also applicable for semirings (without using the commutativity of the addition given per definition of a semiring), see srgcom4lem 20108. (Contributed by GΓ©rard Lang, 4-Dec-2014.) Variant of rglcom4d 20106 for rings. (Revised by AV, 5-Feb-2025.)
𝐡 = (Baseβ€˜π‘…)    &    + = (+gβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((𝑋 + 𝑋) + (π‘Œ + π‘Œ)) = ((𝑋 + π‘Œ) + (𝑋 + π‘Œ)))
 
Theoremringcom 20169 Commutativity of the additive group of a ring. (See also lmodcom 20663.) This proof requires the existence of a multiplicative identity, and the existence of additive inverses. Therefore, this proof is not applicable for semirings. (Contributed by GΓ©rard Lang, 4-Dec-2014.) (Proof shortened by AV, 1-Feb-2025.)
𝐡 = (Baseβ€˜π‘…)    &    + = (+gβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 + π‘Œ) = (π‘Œ + 𝑋))
 
Theoremringabl 20170 A ring is an Abelian group. (Contributed by NM, 26-Aug-2011.)
(𝑅 ∈ Ring β†’ 𝑅 ∈ Abel)
 
Theoremringcmn 20171 A ring is a commutative monoid. (Contributed by Mario Carneiro, 7-Jan-2015.)
(𝑅 ∈ Ring β†’ 𝑅 ∈ CMnd)
 
Theoremringabld 20172 A ring is an Abelian group. (Contributed by SN, 1-Jun-2024.)
(πœ‘ β†’ 𝑅 ∈ Ring)    β‡’   (πœ‘ β†’ 𝑅 ∈ Abel)
 
Theoremringcmnd 20173 A ring is a commutative monoid. (Contributed by SN, 1-Jun-2024.)
(πœ‘ β†’ 𝑅 ∈ Ring)    β‡’   (πœ‘ β†’ 𝑅 ∈ CMnd)
 
Theoremringrng 20174 A unital ring is a non-unital ring. (Contributed by AV, 6-Jan-2020.)
(𝑅 ∈ Ring β†’ 𝑅 ∈ Rng)
 
Theoremringssrng 20175 The unital rings are non-unital rings. (Contributed by AV, 20-Mar-2020.)
Ring βŠ† Rng
 
Theoremisringrng 20176* The predicate "is a unital ring" as extension of the predicate "is a non-unital ring". (Contributed by AV, 17-Feb-2020.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    β‡’   (𝑅 ∈ Ring ↔ (𝑅 ∈ Rng ∧ βˆƒπ‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 ((π‘₯ Β· 𝑦) = 𝑦 ∧ (𝑦 Β· π‘₯) = 𝑦)))
 
Theoremringpropd 20177* If two structures have the same base set, and the values of their group (addition) and ring (multiplication) operations are equal for all pairs of elements of the base set, one is a ring iff the other one is. (Contributed by Mario Carneiro, 6-Dec-2014.) (Revised by Mario Carneiro, 6-Jan-2015.)
(πœ‘ β†’ 𝐡 = (Baseβ€˜πΎ))    &   (πœ‘ β†’ 𝐡 = (Baseβ€˜πΏ))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯(+gβ€˜πΎ)𝑦) = (π‘₯(+gβ€˜πΏ)𝑦))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯(.rβ€˜πΎ)𝑦) = (π‘₯(.rβ€˜πΏ)𝑦))    β‡’   (πœ‘ β†’ (𝐾 ∈ Ring ↔ 𝐿 ∈ Ring))
 
Theoremcrngpropd 20178* If two structures have the same group components (properties), one is a commutative ring iff the other one is. (Contributed by Mario Carneiro, 8-Feb-2015.)
(πœ‘ β†’ 𝐡 = (Baseβ€˜πΎ))    &   (πœ‘ β†’ 𝐡 = (Baseβ€˜πΏ))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯(+gβ€˜πΎ)𝑦) = (π‘₯(+gβ€˜πΏ)𝑦))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯(.rβ€˜πΎ)𝑦) = (π‘₯(.rβ€˜πΏ)𝑦))    β‡’   (πœ‘ β†’ (𝐾 ∈ CRing ↔ 𝐿 ∈ CRing))
 
Theoremringprop 20179 If two structures have the same ring components (properties), one is a ring iff the other one is. (Contributed by Mario Carneiro, 11-Oct-2013.)
(Baseβ€˜πΎ) = (Baseβ€˜πΏ)    &   (+gβ€˜πΎ) = (+gβ€˜πΏ)    &   (.rβ€˜πΎ) = (.rβ€˜πΏ)    β‡’   (𝐾 ∈ Ring ↔ 𝐿 ∈ Ring)
 
Theoremisringd 20180* Properties that determine a ring. (Contributed by NM, 2-Aug-2013.)
(πœ‘ β†’ 𝐡 = (Baseβ€˜π‘…))    &   (πœ‘ β†’ + = (+gβ€˜π‘…))    &   (πœ‘ β†’ Β· = (.rβ€˜π‘…))    &   (πœ‘ β†’ 𝑅 ∈ Grp)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡) β†’ (π‘₯ Β· 𝑦) ∈ 𝐡)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡)) β†’ ((π‘₯ Β· 𝑦) Β· 𝑧) = (π‘₯ Β· (𝑦 Β· 𝑧)))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡)) β†’ (π‘₯ Β· (𝑦 + 𝑧)) = ((π‘₯ Β· 𝑦) + (π‘₯ Β· 𝑧)))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡)) β†’ ((π‘₯ + 𝑦) Β· 𝑧) = ((π‘₯ Β· 𝑧) + (𝑦 Β· 𝑧)))    &   (πœ‘ β†’ 1 ∈ 𝐡)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐡) β†’ ( 1 Β· π‘₯) = π‘₯)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐡) β†’ (π‘₯ Β· 1 ) = π‘₯)    β‡’   (πœ‘ β†’ 𝑅 ∈ Ring)
 
Theoremiscrngd 20181* Properties that determine a commutative ring. (Contributed by Mario Carneiro, 7-Jan-2015.)
(πœ‘ β†’ 𝐡 = (Baseβ€˜π‘…))    &   (πœ‘ β†’ + = (+gβ€˜π‘…))    &   (πœ‘ β†’ Β· = (.rβ€˜π‘…))    &   (πœ‘ β†’ 𝑅 ∈ Grp)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡) β†’ (π‘₯ Β· 𝑦) ∈ 𝐡)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡)) β†’ ((π‘₯ Β· 𝑦) Β· 𝑧) = (π‘₯ Β· (𝑦 Β· 𝑧)))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡)) β†’ (π‘₯ Β· (𝑦 + 𝑧)) = ((π‘₯ Β· 𝑦) + (π‘₯ Β· 𝑧)))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡)) β†’ ((π‘₯ + 𝑦) Β· 𝑧) = ((π‘₯ Β· 𝑧) + (𝑦 Β· 𝑧)))    &   (πœ‘ β†’ 1 ∈ 𝐡)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐡) β†’ ( 1 Β· π‘₯) = π‘₯)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐡) β†’ (π‘₯ Β· 1 ) = π‘₯)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡) β†’ (π‘₯ Β· 𝑦) = (𝑦 Β· π‘₯))    β‡’   (πœ‘ β†’ 𝑅 ∈ CRing)
 
Theoremringlz 20182 The zero of a unital ring is a left-absorbing element. (Contributed by FL, 31-Aug-2009.) (Proof shortened by AV, 30-Mar-2025.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐡) β†’ ( 0 Β· 𝑋) = 0 )
 
Theoremringrz 20183 The zero of a unital ring is a right-absorbing element. (Contributed by FL, 31-Aug-2009.) (Proof shortened by AV, 30-Mar-2025.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐡) β†’ (𝑋 Β· 0 ) = 0 )
 
Theoremringlzd 20184 The zero of a unital ring is a left-absorbing element. (Contributed by SN, 7-Mar-2025.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    β‡’   (πœ‘ β†’ ( 0 Β· 𝑋) = 0 )
 
Theoremringrzd 20185 The zero of a unital ring is a right-absorbing element. (Contributed by SN, 7-Mar-2025.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    β‡’   (πœ‘ β†’ (𝑋 Β· 0 ) = 0 )
 
Theoremringsrg 20186 Any ring is also a semiring. (Contributed by Thierry Arnoux, 1-Apr-2018.)
(𝑅 ∈ Ring β†’ 𝑅 ∈ SRing)
 
Theoremring1eq0 20187 If one and zero are equal, then any two elements of a ring are equal. Alternately, every ring has one distinct from zero except the zero ring containing the single element {0}. (Contributed by Mario Carneiro, 10-Sep-2014.)
𝐡 = (Baseβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ( 1 = 0 β†’ 𝑋 = π‘Œ))
 
Theoremring1ne0 20188 If a ring has at least two elements, its one and zero are different. (Contributed by AV, 13-Apr-2019.)
𝐡 = (Baseβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 1 < (β™―β€˜π΅)) β†’ 1 β‰  0 )
 
Theoremringinvnz1ne0 20189* In a unital ring, a left invertible element is different from zero iff 1 β‰  0. (Contributed by FL, 18-Apr-2010.) (Revised by AV, 24-Aug-2021.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ βˆƒπ‘Ž ∈ 𝐡 (π‘Ž Β· 𝑋) = 1 )    β‡’   (πœ‘ β†’ (𝑋 β‰  0 ↔ 1 β‰  0 ))
 
Theoremringinvnzdiv 20190* In a unital ring, a left invertible element is not a zero divisor. (Contributed by FL, 18-Apr-2010.) (Revised by Jeff Madsen, 18-Apr-2010.) (Revised by AV, 24-Aug-2021.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ βˆƒπ‘Ž ∈ 𝐡 (π‘Ž Β· 𝑋) = 1 )    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    β‡’   (πœ‘ β†’ ((𝑋 Β· π‘Œ) = 0 ↔ π‘Œ = 0 ))
 
Theoremringnegl 20191 Negation in a ring is the same as left multiplication by -1. (rngonegmn1l 37113 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    &   π‘ = (invgβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    β‡’   (πœ‘ β†’ ((π‘β€˜ 1 ) Β· 𝑋) = (π‘β€˜π‘‹))
 
Theoremringnegr 20192 Negation in a ring is the same as right multiplication by -1. (rngonegmn1r 37114 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    &   π‘ = (invgβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    β‡’   (πœ‘ β†’ (𝑋 Β· (π‘β€˜ 1 )) = (π‘β€˜π‘‹))
 
Theoremringmneg1 20193 Negation of a product in a ring. (mulneg1 11655 analog.) Compared with rngmneg1 20062, the proof is shorter making use of the existence of a ring unity. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &   π‘ = (invgβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    β‡’   (πœ‘ β†’ ((π‘β€˜π‘‹) Β· π‘Œ) = (π‘β€˜(𝑋 Β· π‘Œ)))
 
Theoremringmneg2 20194 Negation of a product in a ring. (mulneg2 11656 analog.) Compared with rngmneg2 20063, the proof is shorter making use of the existence of a ring unity. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &   π‘ = (invgβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    β‡’   (πœ‘ β†’ (𝑋 Β· (π‘β€˜π‘Œ)) = (π‘β€˜(𝑋 Β· π‘Œ)))
 
Theoremringm2neg 20195 Double negation of a product in a ring. (mul2neg 11658 analog.) (Contributed by Mario Carneiro, 4-Dec-2014.) (Proof shortened by AV, 30-Mar-2025.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &   π‘ = (invgβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    β‡’   (πœ‘ β†’ ((π‘β€˜π‘‹) Β· (π‘β€˜π‘Œ)) = (𝑋 Β· π‘Œ))
 
Theoremringsubdi 20196 Ring multiplication distributes over subtraction. (subdi 11652 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    βˆ’ = (-gβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    &   (πœ‘ β†’ 𝑍 ∈ 𝐡)    β‡’   (πœ‘ β†’ (𝑋 Β· (π‘Œ βˆ’ 𝑍)) = ((𝑋 Β· π‘Œ) βˆ’ (𝑋 Β· 𝑍)))
 
Theoremringsubdir 20197 Ring multiplication distributes over subtraction. (subdir 11653 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    βˆ’ = (-gβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    &   (πœ‘ β†’ 𝑍 ∈ 𝐡)    β‡’   (πœ‘ β†’ ((𝑋 βˆ’ π‘Œ) Β· 𝑍) = ((𝑋 Β· 𝑍) βˆ’ (π‘Œ Β· 𝑍)))
 
Theoremmulgass2 20198 An associative property between group multiple and ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.gβ€˜π‘…)    &    Γ— = (.rβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ (𝑁 ∈ β„€ ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ ((𝑁 Β· 𝑋) Γ— π‘Œ) = (𝑁 Β· (𝑋 Γ— π‘Œ)))
 
Theoremring1 20199 The (smallest) structure representing a zero ring. (Contributed by AV, 28-Apr-2019.)
𝑀 = {⟨(Baseβ€˜ndx), {𝑍}⟩, ⟨(+gβ€˜ndx), {βŸ¨βŸ¨π‘, π‘βŸ©, π‘βŸ©}⟩, ⟨(.rβ€˜ndx), {βŸ¨βŸ¨π‘, π‘βŸ©, π‘βŸ©}⟩}    β‡’   (𝑍 ∈ 𝑉 β†’ 𝑀 ∈ Ring)
 
Theoremringn0 20200 Rings exist. (Contributed by AV, 29-Apr-2019.)
Ring β‰  βˆ…
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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 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330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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