Home Metamath Proof ExplorerTheorem List (p. 193 of 424) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-27745) Hilbert Space Explorer (27746-29270) Users' Mathboxes (29271-42316)

Theorem List for Metamath Proof Explorer - 19201-19300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlidl1 19201 Every ring contains a unit ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝑈 = (LIdeal‘𝑅)    &   𝐵 = (Base‘𝑅)       (𝑅 ∈ Ring → 𝐵𝑈)

Theoremlidlacs 19202 The ideal system is an algebraic closure system on the base set. (Contributed by Stefan O'Rear, 4-Apr-2015.)
𝐵 = (Base‘𝑊)    &   𝐼 = (LIdeal‘𝑊)       (𝑊 ∈ Ring → 𝐼 ∈ (ACS‘𝐵))

Theoremrspcl 19203 The span of a set of ring elements is an ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
𝐾 = (RSpan‘𝑅)    &   𝐵 = (Base‘𝑅)    &   𝑈 = (LIdeal‘𝑅)       ((𝑅 ∈ Ring ∧ 𝐺𝐵) → (𝐾𝐺) ∈ 𝑈)

Theoremrspssid 19204 The span of a set of ring elements contains those elements. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝐾 = (RSpan‘𝑅)    &   𝐵 = (Base‘𝑅)       ((𝑅 ∈ Ring ∧ 𝐺𝐵) → 𝐺 ⊆ (𝐾𝐺))

Theoremrsp1 19205 The span of the identity element is the unit ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝐾 = (RSpan‘𝑅)    &   𝐵 = (Base‘𝑅)    &    1 = (1r𝑅)       (𝑅 ∈ Ring → (𝐾‘{ 1 }) = 𝐵)

Theoremrsp0 19206 The span of the zero element is the zero ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝐾 = (RSpan‘𝑅)    &    0 = (0g𝑅)       (𝑅 ∈ Ring → (𝐾‘{ 0 }) = { 0 })

Theoremrspssp 19207 The ideal span of a set of elements in a ring is contained in any subring which contains those elements. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝐾 = (RSpan‘𝑅)    &   𝑈 = (LIdeal‘𝑅)       ((𝑅 ∈ Ring ∧ 𝐼𝑈𝐺𝐼) → (𝐾𝐺) ⊆ 𝐼)

Theoremmrcrsp 19208 Moore closure generalizes ideal span. (Contributed by Stefan O'Rear, 4-Apr-2015.)
𝑈 = (LIdeal‘𝑅)    &   𝐾 = (RSpan‘𝑅)    &   𝐹 = (mrCls‘𝑈)       (𝑅 ∈ Ring → 𝐾 = 𝐹)

Theoremlidlnz 19209* A nonzero ideal contains a nonzero element. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝑈 = (LIdeal‘𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ Ring ∧ 𝐼𝑈𝐼 ≠ { 0 }) → ∃𝑥𝐼 𝑥0 )

Theoremdrngnidl 19210 A division ring has only the two trivial ideals. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Wolf Lammen, 6-Sep-2020.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝑈 = (LIdeal‘𝑅)       (𝑅 ∈ DivRing → 𝑈 = {{ 0 }, 𝐵})

Theoremlidlrsppropd 19211* The left ideals and ring span of a ring depend only on the ring components. Here 𝑊 is expected to be either 𝐵 (when closure is available) or V (when strong equality is available). (Contributed by Mario Carneiro, 14-Jun-2015.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   (𝜑𝐵𝑊)    &   ((𝜑 ∧ (𝑥𝑊𝑦𝑊)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(.r𝐾)𝑦) ∈ 𝑊)    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(.r𝐾)𝑦) = (𝑥(.r𝐿)𝑦))       (𝜑 → ((LIdeal‘𝐾) = (LIdeal‘𝐿) ∧ (RSpan‘𝐾) = (RSpan‘𝐿)))

10.8.2  Two-sided ideals and quotient rings

Syntaxc2idl 19212 Ring two-sided ideal function.
class 2Ideal

Definitiondf-2idl 19213 Define the class of two-sided ideals of a ring. A two-sided ideal is a left ideal which is also a right ideal (or a left ideal over the opposite ring). (Contributed by Mario Carneiro, 14-Jun-2015.)
2Ideal = (𝑟 ∈ V ↦ ((LIdeal‘𝑟) ∩ (LIdeal‘(oppr𝑟))))

Theorem2idlval 19214 Definition of a two-sided ideal. (Contributed by Mario Carneiro, 14-Jun-2015.)
𝐼 = (LIdeal‘𝑅)    &   𝑂 = (oppr𝑅)    &   𝐽 = (LIdeal‘𝑂)    &   𝑇 = (2Ideal‘𝑅)       𝑇 = (𝐼𝐽)

Theorem2idlcpbl 19215 The coset equivalence relation for a two-sided ideal is compatible with ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015.)
𝑋 = (Base‘𝑅)    &   𝐸 = (𝑅 ~QG 𝑆)    &   𝐼 = (2Ideal‘𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ Ring ∧ 𝑆𝐼) → ((𝐴𝐸𝐶𝐵𝐸𝐷) → (𝐴 · 𝐵)𝐸(𝐶 · 𝐷)))

Theoremqus1 19216 The multiplicative identity of the quotient ring. (Contributed by Mario Carneiro, 14-Jun-2015.)
𝑈 = (𝑅 /s (𝑅 ~QG 𝑆))    &   𝐼 = (2Ideal‘𝑅)    &    1 = (1r𝑅)       ((𝑅 ∈ Ring ∧ 𝑆𝐼) → (𝑈 ∈ Ring ∧ [ 1 ](𝑅 ~QG 𝑆) = (1r𝑈)))

Theoremqusring 19217 If 𝑆 is a two-sided ideal in 𝑅, then 𝑈 = 𝑅 / 𝑆 is a ring, called the quotient ring of 𝑅 by 𝑆. (Contributed by Mario Carneiro, 14-Jun-2015.)
𝑈 = (𝑅 /s (𝑅 ~QG 𝑆))    &   𝐼 = (2Ideal‘𝑅)       ((𝑅 ∈ Ring ∧ 𝑆𝐼) → 𝑈 ∈ Ring)

Theoremqusrhm 19218* If 𝑆 is a two-sided ideal in 𝑅, then the "natural map" from elements to their cosets is a ring homomorphism from 𝑅 to 𝑅 / 𝑆. (Contributed by Mario Carneiro, 15-Jun-2015.)
𝑈 = (𝑅 /s (𝑅 ~QG 𝑆))    &   𝐼 = (2Ideal‘𝑅)    &   𝑋 = (Base‘𝑅)    &   𝐹 = (𝑥𝑋 ↦ [𝑥](𝑅 ~QG 𝑆))       ((𝑅 ∈ Ring ∧ 𝑆𝐼) → 𝐹 ∈ (𝑅 RingHom 𝑈))

Theoremcrngridl 19219 In a commutative ring, the left and right ideals coincide. (Contributed by Mario Carneiro, 14-Jun-2015.)
𝐼 = (LIdeal‘𝑅)    &   𝑂 = (oppr𝑅)       (𝑅 ∈ CRing → 𝐼 = (LIdeal‘𝑂))

Theoremcrng2idl 19220 In a commutative ring, a two-sided ideal is the same as a left ideal. (Contributed by Mario Carneiro, 14-Jun-2015.)
𝐼 = (LIdeal‘𝑅)       (𝑅 ∈ CRing → 𝐼 = (2Ideal‘𝑅))

Theoremquscrng 19221 The quotient of a commutative ring by an ideal is a commutative ring. (Contributed by Mario Carneiro, 15-Jun-2015.)
𝑈 = (𝑅 /s (𝑅 ~QG 𝑆))    &   𝐼 = (LIdeal‘𝑅)       ((𝑅 ∈ CRing ∧ 𝑆𝐼) → 𝑈 ∈ CRing)

10.8.3  Principal ideal rings. Divisibility in the integers

Syntaxclpidl 19222 Ring left-principal-ideal function.
class LPIdeal

Syntaxclpir 19223 Class of left principal ideal rings.
class LPIR

Definitiondf-lpidl 19224* Define the class of left principal ideals of a ring, which are ideals with a single generator. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LPIdeal = (𝑤 ∈ Ring ↦ 𝑔 ∈ (Base‘𝑤){((RSpan‘𝑤)‘{𝑔})})

Definitiondf-lpir 19225 Define the class of left principal ideal rings, rings where every left ideal has a single generator. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LPIR = {𝑤 ∈ Ring ∣ (LIdeal‘𝑤) = (LPIdeal‘𝑤)}

Theoremlpival 19226* Value of the set of principal ideals. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝑃 = (LPIdeal‘𝑅)    &   𝐾 = (RSpan‘𝑅)    &   𝐵 = (Base‘𝑅)       (𝑅 ∈ Ring → 𝑃 = 𝑔𝐵 {(𝐾‘{𝑔})})

Theoremislpidl 19227* Property of being a principal ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝑃 = (LPIdeal‘𝑅)    &   𝐾 = (RSpan‘𝑅)    &   𝐵 = (Base‘𝑅)       (𝑅 ∈ Ring → (𝐼𝑃 ↔ ∃𝑔𝐵 𝐼 = (𝐾‘{𝑔})))

Theoremlpi0 19228 The zero ideal is always principal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝑃 = (LPIdeal‘𝑅)    &    0 = (0g𝑅)       (𝑅 ∈ Ring → { 0 } ∈ 𝑃)

Theoremlpi1 19229 The unit ideal is always principal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝑃 = (LPIdeal‘𝑅)    &   𝐵 = (Base‘𝑅)       (𝑅 ∈ Ring → 𝐵𝑃)

Theoremislpir 19230 Principal ideal rings are where all ideals are principal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝑃 = (LPIdeal‘𝑅)    &   𝑈 = (LIdeal‘𝑅)       (𝑅 ∈ LPIR ↔ (𝑅 ∈ Ring ∧ 𝑈 = 𝑃))

Theoremlpiss 19231 Principal ideals are a subclass of ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝑃 = (LPIdeal‘𝑅)    &   𝑈 = (LIdeal‘𝑅)       (𝑅 ∈ Ring → 𝑃𝑈)

Theoremislpir2 19232 Principal ideal rings are where all ideals are principal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝑃 = (LPIdeal‘𝑅)    &   𝑈 = (LIdeal‘𝑅)       (𝑅 ∈ LPIR ↔ (𝑅 ∈ Ring ∧ 𝑈𝑃))

Theoremlpirring 19233 Principal ideal rings are rings. (Contributed by Stefan O'Rear, 24-Jan-2015.)
(𝑅 ∈ LPIR → 𝑅 ∈ Ring)

Theoremdrnglpir 19234 Division rings are principal ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
(𝑅 ∈ DivRing → 𝑅 ∈ LPIR)

Theoremrspsn 19235* Membership in principal ideals is closely related to divisibility. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.)
𝐵 = (Base‘𝑅)    &   𝐾 = (RSpan‘𝑅)    &    = (∥r𝑅)       ((𝑅 ∈ Ring ∧ 𝐺𝐵) → (𝐾‘{𝐺}) = {𝑥𝐺 𝑥})

Theoremlidldvgen 19236* An element generates an ideal iff it is contained in the ideal and all elements are right-divided by it. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝐵 = (Base‘𝑅)    &   𝑈 = (LIdeal‘𝑅)    &   𝐾 = (RSpan‘𝑅)    &    = (∥r𝑅)       ((𝑅 ∈ Ring ∧ 𝐼𝑈𝐺𝐵) → (𝐼 = (𝐾‘{𝐺}) ↔ (𝐺𝐼 ∧ ∀𝑥𝐼 𝐺 𝑥)))

Theoremlpigen 19237* An ideal is principal iff it contains an element which right-divides all elements. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Wolf Lammen, 6-Sep-2020.)
𝑈 = (LIdeal‘𝑅)    &   𝑃 = (LPIdeal‘𝑅)    &    = (∥r𝑅)       ((𝑅 ∈ Ring ∧ 𝐼𝑈) → (𝐼𝑃 ↔ ∃𝑥𝐼𝑦𝐼 𝑥 𝑦))

10.8.4  Nonzero rings and zero rings

Syntaxcnzr 19238 The class of nonzero rings.
class NzRing

Definitiondf-nzr 19239 A nonzero or nontrivial ring is a ring with at least two values, or equivalently where 1 and 0 are different. (Contributed by Stefan O'Rear, 24-Feb-2015.)
NzRing = {𝑟 ∈ Ring ∣ (1r𝑟) ≠ (0g𝑟)}

Theoremisnzr 19240 Property of a nonzero ring. (Contributed by Stefan O'Rear, 24-Feb-2015.)
1 = (1r𝑅)    &    0 = (0g𝑅)       (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 10 ))

Theoremnzrnz 19241 One and zero are different in a nonzero ring. (Contributed by Stefan O'Rear, 24-Feb-2015.)
1 = (1r𝑅)    &    0 = (0g𝑅)       (𝑅 ∈ NzRing → 10 )

Theoremnzrring 19242 A nonzero ring is a ring. (Contributed by Stefan O'Rear, 24-Feb-2015.)
(𝑅 ∈ NzRing → 𝑅 ∈ Ring)

Theoremdrngnzr 19243 All division rings are nonzero. (Contributed by Stefan O'Rear, 24-Feb-2015.)
(𝑅 ∈ DivRing → 𝑅 ∈ NzRing)

Theoremisnzr2 19244 Equivalent characterization of nonzero rings: they have at least two elements. (Contributed by Stefan O'Rear, 24-Feb-2015.)
𝐵 = (Base‘𝑅)       (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 2𝑜𝐵))

Theoremisnzr2hash 19245 Equivalent characterization of nonzero rings: they have at least two elements. Analogous to isnzr2 19244. (Contributed by AV, 14-Apr-2019.)
𝐵 = (Base‘𝑅)       (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 1 < (#‘𝐵)))

Theoremopprnzr 19246 The opposite of a nonzero ring is nonzero. (Contributed by Mario Carneiro, 17-Jun-2015.)
𝑂 = (oppr𝑅)       (𝑅 ∈ NzRing → 𝑂 ∈ NzRing)

Theoremringelnzr 19247 A ring is nonzero if it has a nonzero element. (Contributed by Stefan O'Rear, 6-Feb-2015.) (Revised by Mario Carneiro, 13-Jun-2015.)
0 = (0g𝑅)    &   𝐵 = (Base‘𝑅)       ((𝑅 ∈ Ring ∧ 𝑋 ∈ (𝐵 ∖ { 0 })) → 𝑅 ∈ NzRing)

Theoremnzrunit 19248 A unit is nonzero in any nonzero ring. (Contributed by Mario Carneiro, 6-Oct-2015.)
𝑈 = (Unit‘𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ NzRing ∧ 𝐴𝑈) → 𝐴0 )

Theoremsubrgnzr 19249 A subring of a nonzero ring is nonzero. (Contributed by Mario Carneiro, 15-Jun-2015.)
𝑆 = (𝑅s 𝐴)       ((𝑅 ∈ NzRing ∧ 𝐴 ∈ (SubRing‘𝑅)) → 𝑆 ∈ NzRing)

Theorem0ringnnzr 19250 A ring is a zero ring iff it is not a nonzero ring. (Contributed by AV, 14-Apr-2019.)
(𝑅 ∈ Ring → ((#‘(Base‘𝑅)) = 1 ↔ ¬ 𝑅 ∈ NzRing))

Theorem0ring 19251 If a ring has only one element, it is the zero ring. According to Wikipedia ("Zero ring", 14-Apr-2019, https://en.wikipedia.org/wiki/Zero_ring): "The zero ring, denoted {0} or simply 0, consists of the one-element set {0} with the operations + and * defined so that 0 + 0 = 0 and 0 * 0 = 0.". (Contributed by AV, 14-Apr-2019.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ Ring ∧ (#‘𝐵) = 1) → 𝐵 = { 0 })

Theorem0ring01eq 19252 In a ring with only one element, i.e. a zero ring, the zero and the identity element are the same. (Contributed by AV, 14-Apr-2019.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    1 = (1r𝑅)       ((𝑅 ∈ Ring ∧ (#‘𝐵) = 1) → 0 = 1 )

Theorem01eq0ring 19253 If the zero and the identity element of a ring are the same, the ring is the zero ring. (Contributed by AV, 16-Apr-2019.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    1 = (1r𝑅)       ((𝑅 ∈ Ring ∧ 0 = 1 ) → 𝐵 = { 0 })

Theorem0ring01eqbi 19254 In a unital ring the zero equals the unity iff the ring is the zero ring. (Contributed by FL, 14-Feb-2010.) (Revised by AV, 23-Jan-2020.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    1 = (1r𝑅)       (𝑅 ∈ Ring → (𝐵 ≈ 1𝑜1 = 0 ))

Theoremrng1nnzr 19255 The (smallest) structure representing a zero ring is not a nonzero ring. (Contributed by AV, 29-Apr-2019.)
𝑀 = {⟨(Base‘ndx), {𝑍}⟩, ⟨(+g‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩, ⟨(.r‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩}       (𝑍𝑉𝑀 ∉ NzRing)

Theoremring1zr 19256 The only (unital) ring with a base set consisting of one element is the zero ring (at least if its operations are internal binary operations). Note: The assumption 𝑅 ∈ Ring could be weakened if a definition of a non-unital ring ("Rng") was available (it would be sufficient that the multiplication is closed). (Contributed by FL, 13-Feb-2010.) (Revised by AV, 25-Jan-2020.) (Proof shortened by AV, 7-Feb-2020.)
𝐵 = (Base‘𝑅)    &    + = (+g𝑅)    &    = (.r𝑅)       (((𝑅 ∈ Ring ∧ + Fn (𝐵 × 𝐵) ∧ Fn (𝐵 × 𝐵)) ∧ 𝑍𝐵) → (𝐵 = {𝑍} ↔ ( + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∧ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩})))

Theoremrngen1zr 19257 The only (unital) ring with one element is the zero ring (at least if its operations are internal binary operations). Note: The assumption 𝑅 ∈ Ring could be weakened if a definition of a non-unital ring ("Rng") was available (it would be sufficient that the multiplication is closed). (Contributed by FL, 14-Feb-2010.) (Revised by AV, 25-Jan-2020.)
𝐵 = (Base‘𝑅)    &    + = (+g𝑅)    &    = (.r𝑅)       (((𝑅 ∈ Ring ∧ + Fn (𝐵 × 𝐵) ∧ Fn (𝐵 × 𝐵)) ∧ 𝑍𝐵) → (𝐵 ≈ 1𝑜 ↔ ( + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∧ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩})))

Theoremringen1zr 19258 The only unital ring with one element is the zero ring (at least if its operations are internal binary operations). Note: The assumption 𝑅 ∈ Ring could be weakened if a definition of a non-unital ring ("Rng") was available (it would be sufficient that the multiplication is closed). (Contributed by FL, 15-Feb-2010.) (Revised by AV, 25-Jan-2020.)
𝐵 = (Base‘𝑅)    &    + = (+g𝑅)    &    = (.r𝑅)    &   𝑍 = (0g𝑅)       ((𝑅 ∈ Ring ∧ + Fn (𝐵 × 𝐵) ∧ Fn (𝐵 × 𝐵)) → (𝐵 ≈ 1𝑜 ↔ ( + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∧ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩})))

Theoremrng1nfld 19259 The zero ring is not a field. (Contributed by AV, 29-Apr-2019.)
𝑀 = {⟨(Base‘ndx), {𝑍}⟩, ⟨(+g‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩, ⟨(.r‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩}       (𝑍𝑉𝑀 ∉ Field)

10.8.5  Left regular elements. More kinds of rings

Syntaxcrlreg 19260 Set of left-regular elements in a ring.
class RLReg

Syntaxcdomn 19261 Class of (ring theoretic) domains.
class Domn

Syntaxcidom 19262 Class of integral domains.
class IDomn

Syntaxcpid 19263 Class of principal ideal domains.
class PID

Definitiondf-rlreg 19264* Define the set of left-regular elements in a ring as those elements which are not left zero divisors, meaning that multiplying a nonzero element on the left by a left-regular element gives a nonzero product. (Contributed by Stefan O'Rear, 22-Mar-2015.)
RLReg = (𝑟 ∈ V ↦ {𝑥 ∈ (Base‘𝑟) ∣ ∀𝑦 ∈ (Base‘𝑟)((𝑥(.r𝑟)𝑦) = (0g𝑟) → 𝑦 = (0g𝑟))})

Definitiondf-domn 19265* A domain is a nonzero ring in which there are no nontrivial zero divisors. (Contributed by Mario Carneiro, 28-Mar-2015.)
Domn = {𝑟 ∈ NzRing ∣ [(Base‘𝑟) / 𝑏][(0g𝑟) / 𝑧]𝑥𝑏𝑦𝑏 ((𝑥(.r𝑟)𝑦) = 𝑧 → (𝑥 = 𝑧𝑦 = 𝑧))}

Definitiondf-idom 19266 An integral domain is a commutative domain. (Contributed by Mario Carneiro, 17-Jun-2015.)
IDomn = (CRing ∩ Domn)

Definitiondf-pid 19267 A principal ideal domain is an integral domain satisfying the left principal ideal property. (Contributed by Stefan O'Rear, 29-Mar-2015.)
PID = (IDomn ∩ LPIR)

Theoremrrgval 19268* Value of the set or left-regular elements in a ring. (Contributed by Stefan O'Rear, 22-Mar-2015.)
𝐸 = (RLReg‘𝑅)    &   𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    0 = (0g𝑅)       𝐸 = {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )}

Theoremisrrg 19269* Membership in the set of left-regular elements. (Contributed by Stefan O'Rear, 22-Mar-2015.)
𝐸 = (RLReg‘𝑅)    &   𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    0 = (0g𝑅)       (𝑋𝐸 ↔ (𝑋𝐵 ∧ ∀𝑦𝐵 ((𝑋 · 𝑦) = 0𝑦 = 0 )))

Theoremrrgeq0i 19270 Property of a left-regular element. (Contributed by Stefan O'Rear, 22-Mar-2015.)
𝐸 = (RLReg‘𝑅)    &   𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    0 = (0g𝑅)       ((𝑋𝐸𝑌𝐵) → ((𝑋 · 𝑌) = 0𝑌 = 0 ))

Theoremrrgeq0 19271 Left-multiplication by a left regular element does not change zeroness. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝐸 = (RLReg‘𝑅)    &   𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐸𝑌𝐵) → ((𝑋 · 𝑌) = 0𝑌 = 0 ))

Theoremrrgsupp 19272 Left multiplication by a left regular element does not change the support set of a vector. (Contributed by Stefan O'Rear, 28-Mar-2015.) (Revised by AV, 20-Jul-2019.)
𝐸 = (RLReg‘𝑅)    &   𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    0 = (0g𝑅)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐸)    &   (𝜑𝑌:𝐼𝐵)       (𝜑 → (((𝐼 × {𝑋}) ∘𝑓 · 𝑌) supp 0 ) = (𝑌 supp 0 ))

Theoremrrgss 19273 Left-regular elements are a subset of the base set. (Contributed by Stefan O'Rear, 22-Mar-2015.)
𝐸 = (RLReg‘𝑅)    &   𝐵 = (Base‘𝑅)       𝐸𝐵

Theoremunitrrg 19274 Units are regular elements. (Contributed by Stefan O'Rear, 22-Mar-2015.)
𝐸 = (RLReg‘𝑅)    &   𝑈 = (Unit‘𝑅)       (𝑅 ∈ Ring → 𝑈𝐸)

Theoremisdomn 19275* Expand definition of a domain. (Contributed by Mario Carneiro, 28-Mar-2015.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    0 = (0g𝑅)       (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ ∀𝑥𝐵𝑦𝐵 ((𝑥 · 𝑦) = 0 → (𝑥 = 0𝑦 = 0 ))))

Theoremdomnnzr 19276 A domain is a nonzero ring. (Contributed by Mario Carneiro, 28-Mar-2015.)
(𝑅 ∈ Domn → 𝑅 ∈ NzRing)

Theoremdomnring 19277 A domain is a ring. (Contributed by Mario Carneiro, 28-Mar-2015.)
(𝑅 ∈ Domn → 𝑅 ∈ Ring)

Theoremdomneq0 19278 In a domain, a product is zero iff it has a zero factor. (Contributed by Mario Carneiro, 28-Mar-2015.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ Domn ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 · 𝑌) = 0 ↔ (𝑋 = 0𝑌 = 0 )))

Theoremdomnmuln0 19279 In a domain, a product of nonzero elements is nonzero. (Contributed by Mario Carneiro, 6-May-2015.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ Domn ∧ (𝑋𝐵𝑋0 ) ∧ (𝑌𝐵𝑌0 )) → (𝑋 · 𝑌) ≠ 0 )

Theoremisdomn2 19280 A ring is a domain iff all nonzero elements are nonzero-divisors. (Contributed by Mario Carneiro, 28-Mar-2015.)
𝐵 = (Base‘𝑅)    &   𝐸 = (RLReg‘𝑅)    &    0 = (0g𝑅)       (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ (𝐵 ∖ { 0 }) ⊆ 𝐸))

Theoremdomnrrg 19281 In a domain, any nonzero element is a nonzero-divisor. (Contributed by Mario Carneiro, 28-Mar-2015.)
𝐵 = (Base‘𝑅)    &   𝐸 = (RLReg‘𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ Domn ∧ 𝑋𝐵𝑋0 ) → 𝑋𝐸)

Theoremopprdomn 19282 The opposite of a domain is also a domain. (Contributed by Mario Carneiro, 15-Jun-2015.)
𝑂 = (oppr𝑅)       (𝑅 ∈ Domn → 𝑂 ∈ Domn)

Theoremabvn0b 19283 Another characterization of domains, hinted at in abvtriv 18822: a nonzero ring is a domain iff it has an absolute value. (Contributed by Mario Carneiro, 6-May-2015.)
𝐴 = (AbsVal‘𝑅)       (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ 𝐴 ≠ ∅))

Theoremdrngdomn 19284 A division ring is a domain. (Contributed by Mario Carneiro, 29-Mar-2015.)
(𝑅 ∈ DivRing → 𝑅 ∈ Domn)

Theoremisidom 19285 An integral domain is a commutative domain. (Contributed by Mario Carneiro, 17-Jun-2015.)
(𝑅 ∈ IDomn ↔ (𝑅 ∈ CRing ∧ 𝑅 ∈ Domn))

Theoremfldidom 19286 A field is an integral domain. (Contributed by Mario Carneiro, 29-Mar-2015.)
(𝑅 ∈ Field → 𝑅 ∈ IDomn)

Theoremfidomndrnglem 19287* Lemma for fidomndrng 19288. (Contributed by Mario Carneiro, 15-Jun-2015.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &    = (∥r𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ Domn)    &   (𝜑𝐵 ∈ Fin)    &   (𝜑𝐴 ∈ (𝐵 ∖ { 0 }))    &   𝐹 = (𝑥𝐵 ↦ (𝑥 · 𝐴))       (𝜑𝐴 1 )

Theoremfidomndrng 19288 A finite domain is a division ring. (Contributed by Mario Carneiro, 15-Jun-2015.)
𝐵 = (Base‘𝑅)       (𝐵 ∈ Fin → (𝑅 ∈ Domn ↔ 𝑅 ∈ DivRing))

Theoremfiidomfld 19289 A finite integral domain is a field. (Contributed by Mario Carneiro, 15-Jun-2015.)
𝐵 = (Base‘𝑅)       (𝐵 ∈ Fin → (𝑅 ∈ IDomn ↔ 𝑅 ∈ Field))

10.9  Associative algebras

10.9.1  Definition and basic properties

Syntaxcasa 19290 Associative algebra.
class AssAlg

Syntaxcasp 19291 Algebraic span function.
class AlgSpan

Syntaxcascl 19292 Class of algebra scalar injection function.
class algSc

Definitiondf-assa 19293* Definition of an associative algebra. An associative algebra is a set equipped with a left-module structure on a (commutative) ring, coupled with a multiplicative internal operation on the vectors of the module that is associative and distributive for the additive structure of the left-module (so giving the vectors a ring structure) and that is also bilinear under the scalar product. (Contributed by Mario Carneiro, 29-Dec-2014.)
AssAlg = {𝑤 ∈ (LMod ∩ Ring) ∣ [(Scalar‘𝑤) / 𝑓](𝑓 ∈ CRing ∧ ∀𝑟 ∈ (Base‘𝑓)∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑤)[( ·𝑠𝑤) / 𝑠][(.r𝑤) / 𝑡](((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦))))}

Definitiondf-asp 19294* Define the algebraic span of a set of vectors in an algebra. (Contributed by Mario Carneiro, 7-Jan-2015.)
AlgSpan = (𝑤 ∈ AssAlg ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ {𝑡 ∈ ((SubRing‘𝑤) ∩ (LSubSp‘𝑤)) ∣ 𝑠𝑡}))

Definitiondf-ascl 19295* Every unital algebra contains a canonical homomorphic image of its ring of scalars as scalar multiples of the unit. This names the homomorphism. (Contributed by Mario Carneiro, 8-Mar-2015.)
algSc = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘(Scalar‘𝑤)) ↦ (𝑥( ·𝑠𝑤)(1r𝑤))))

Theoremisassa 19296* The properties of an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐵 = (Base‘𝐹)    &    · = ( ·𝑠𝑊)    &    × = (.r𝑊)       (𝑊 ∈ AssAlg ↔ ((𝑊 ∈ LMod ∧ 𝑊 ∈ Ring ∧ 𝐹 ∈ CRing) ∧ ∀𝑟𝐵𝑥𝑉𝑦𝑉 (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)))))

Theoremassalem 19297 The properties of an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐵 = (Base‘𝐹)    &    · = ( ·𝑠𝑊)    &    × = (.r𝑊)       ((𝑊 ∈ AssAlg ∧ (𝐴𝐵𝑋𝑉𝑌𝑉)) → (((𝐴 · 𝑋) × 𝑌) = (𝐴 · (𝑋 × 𝑌)) ∧ (𝑋 × (𝐴 · 𝑌)) = (𝐴 · (𝑋 × 𝑌))))

Theoremassaass 19298 Left-associative property of an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐵 = (Base‘𝐹)    &    · = ( ·𝑠𝑊)    &    × = (.r𝑊)       ((𝑊 ∈ AssAlg ∧ (𝐴𝐵𝑋𝑉𝑌𝑉)) → ((𝐴 · 𝑋) × 𝑌) = (𝐴 · (𝑋 × 𝑌)))

Theoremassaassr 19299 Right-associative property of an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐵 = (Base‘𝐹)    &    · = ( ·𝑠𝑊)    &    × = (.r𝑊)       ((𝑊 ∈ AssAlg ∧ (𝐴𝐵𝑋𝑉𝑌𝑉)) → (𝑋 × (𝐴 · 𝑌)) = (𝐴 · (𝑋 × 𝑌)))

Theoremassalmod 19300 An associative algebra is a left module. (Contributed by Mario Carneiro, 5-Dec-2014.)
(𝑊 ∈ AssAlg → 𝑊 ∈ LMod)

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 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 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 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 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42316
 Copyright terms: Public domain < Previous  Next >