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Theorem List for Metamath Proof Explorer - 16301-16400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlidlnz 16301* A nonzero ideal contains a nonzero element. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LIdeal

Theoremdrngnidl 16302 A division ring has only the two trivial ideals. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LIdeal

Theoremlidlrsppropd 16303* The left ideals and ring span of a ring depend only on the ring components. Here is expected to be either (when closure is available) or (when strong equality is available). (Contributed by Mario Carneiro, 14-Jun-2015.)
LIdeal LIdeal RSpan RSpan

10.8.2  Two-sided ideals and quotient rings

Syntaxc2idl 16304 Ring two-sided ideal function.
2Ideal

Definitiondf-2idl 16305 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 LIdeal LIdealoppr

Theorem2idlval 16306 Definition of a two-sided ideal. (Contributed by Mario Carneiro, 14-Jun-2015.)
LIdeal       oppr       LIdeal       2Ideal

Theorem2idlcpbl 16307 The coset equivalence relation for a two-sided ideal is compatible with ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015.)
~QG        2Ideal

Theoremdivs1 16308 The multiplicative identity of the quotient ring. (Contributed by Mario Carneiro, 14-Jun-2015.)
s ~QG        2Ideal              ~QG

Theoremdivsrng 16309 If is a two-sided ideal in , then is a ring, called the quotient ring of by . (Contributed by Mario Carneiro, 14-Jun-2015.)
s ~QG        2Ideal

Theoremdivsrhm 16310* If is a two-sided ideal in , then the "natural map" from elements to their cosets is a ring homomorphism from to . (Contributed by Mario Carneiro, 15-Jun-2015.)
s ~QG        2Ideal              ~QG        RingHom

Theoremcrngridl 16311 In a commutative ring, the left and right ideals coincide. (Contributed by Mario Carneiro, 14-Jun-2015.)
LIdeal       oppr       LIdeal

Theoremcrng2idl 16312 In a commutative ring, a two-sided ideal is the same as a left ideal. (Contributed by Mario Carneiro, 14-Jun-2015.)
LIdeal       2Ideal

Theoremdivscrng 16313 The quotient of a commutative ring by an ideal is a commutative ring. (Contributed by Mario Carneiro, 15-Jun-2015.)
s ~QG        LIdeal

10.8.3  Principal ideal rings. Divisibility in the integers

Syntaxclpidl 16314 Ring left-principal-ideal function.
LPIdeal

Syntaxclpir 16315 Class of left principal ideal rings.
LPIR

Definitiondf-lpidl 16316* 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 RSpan

Definitiondf-lpir 16317 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 LIdeal LPIdeal

Theoremlpival 16318* Value of the set of principal ideals. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LPIdeal       RSpan

Theoremislpidl 16319* Property of being a principal ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LPIdeal       RSpan

Theoremlpi0 16320 The zero ideal is always principal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LPIdeal

Theoremlpi1 16321 The unit ideal is always principal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LPIdeal

Theoremislpir 16322 Principal ideal rings are where all ideals are principal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LPIdeal       LIdeal       LPIR

Theoremlpiss 16323 Principal ideals are a subclass of ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LPIdeal       LIdeal

Theoremislpir2 16324 Principal ideal rings are where all ideals are principal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LPIdeal       LIdeal       LPIR

Theoremlpirrng 16325 Principal ideal rings are rings. (Contributed by Stefan O'Rear, 24-Jan-2015.)
LPIR

Theoremdrnglpir 16326 Division rings are principal ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LPIR

Theoremrspsn 16327* Membership in principal ideals is closely related to divisibility. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.)
RSpan       r

Theoremlidldvgen 16328* 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.)
LIdeal       RSpan       r

Theoremlpigen 16329* An ideal is principal iff it contains an element which right-divides all elements. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LIdeal       LPIdeal       r

10.8.4  Nonzero rings

Syntaxcnzr 16330 The class of nonzero rings.
NzRing

Definitiondf-nzr 16331 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

Theoremisnzr 16332 Property of a nonzero ring. (Contributed by Stefan O'Rear, 24-Feb-2015.)
NzRing

Theoremnzrnz 16333 One and zero are different in a nonzero ring. (Contributed by Stefan O'Rear, 24-Feb-2015.)
NzRing

Theoremnzrrng 16334 A nonzero ring is a ring. (Contributed by Stefan O'Rear, 24-Feb-2015.)
NzRing

Theoremdrngnzr 16335 All division rings are nonzero. (Contributed by Stefan O'Rear, 24-Feb-2015.)
NzRing

Theoremisnzr2 16336 Equivalent characterization of nonzero rings: they have at least two elements. (Contributed by Stefan O'Rear, 24-Feb-2015.)
NzRing

Theoremopprnzr 16337 The opposite of a nonzero ring is nonzero. (Contributed by Mario Carneiro, 17-Jun-2015.)
oppr       NzRing NzRing

Theoremrngelnzr 16338 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.)
NzRing

Theoremnzrunit 16339 A unit is nonzero in any nonzero ring. (Contributed by Mario Carneiro, 6-Oct-2015.)
Unit              NzRing

Theoremsubrgnzr 16340 A subring of a nonzero ring is nonzero. (Contributed by Mario Carneiro, 15-Jun-2015.)
s        NzRing SubRing NzRing

10.8.5  Left regular elements. More kinds of rings

Syntaxcrlreg 16341 Set of left-regular elements in a ring.
RLReg

Syntaxcdomn 16342 Class of (ring theoretic) domains.
Domn

Syntaxcidom 16343 Class of integral domains.
IDomn

Syntaxcpid 16344 Class of principal ideal domains.
PID

Definitiondf-rlreg 16345* 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

Definitiondf-domn 16346* A domain is a nonzero ring in which there are no nontrivial zero divisors. (Contributed by Mario Carneiro, 28-Mar-2015.)
Domn NzRing

Definitiondf-idom 16347 An integral domain is a commutative domain. (Contributed by Mario Carneiro, 17-Jun-2015.)
IDomn Domn

Definitiondf-pid 16348 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 16349* Value of the set or left-regular elements in a ring. (Contributed by Stefan O'Rear, 22-Mar-2015.)
RLReg

Theoremisrrg 16350* Membership in the set of left-regular elements. (Contributed by Stefan O'Rear, 22-Mar-2015.)
RLReg

Theoremrrgeq0i 16351 Property of a left-regular element. (Contributed by Stefan O'Rear, 22-Mar-2015.)
RLReg

Theoremrrgeq0 16352 Left-multiplication by a left regular element does not change zeroness. (Contributed by Stefan O'Rear, 28-Mar-2015.)
RLReg

Theoremrrgsupp 16353 Left multiplication by a left regular element does not change the support set of a vector. (Contributed by Stefan O'Rear, 28-Mar-2015.)
RLReg

Theoremrrgss 16354 Left-regular elements are a subset of the base set. (Contributed by Stefan O'Rear, 22-Mar-2015.)
RLReg

Theoremunitrrg 16355 Units are regular elements. (Contributed by Stefan O'Rear, 22-Mar-2015.)
RLReg       Unit

Theoremisdomn 16356* Expand definition of a domain. (Contributed by Mario Carneiro, 28-Mar-2015.)
Domn NzRing

Theoremdomnnzr 16357 A domain is a nonzero ring. (Contributed by Mario Carneiro, 28-Mar-2015.)
Domn NzRing

Theoremdomnrng 16358 A domain is a ring. (Contributed by Mario Carneiro, 28-Mar-2015.)
Domn

Theoremdomneq0 16359 In a domain, a product is zero iff it has a zero factor. (Contributed by Mario Carneiro, 28-Mar-2015.)
Domn

Theoremdomnmuln0 16360 In a domain, a product of nonzero elements is nonzero. (Contributed by Mario Carneiro, 6-May-2015.)
Domn

Theoremisdomn2 16361 A ring is a domain iff all nonzero elements are non-zero-divisors. (Contributed by Mario Carneiro, 28-Mar-2015.)
RLReg              Domn NzRing

Theoremdomnrrg 16362 In a domain, any nonzero element is a non-zero-divisor. (Contributed by Mario Carneiro, 28-Mar-2015.)
RLReg              Domn

Theoremopprdomn 16363 The opposite of a domain is also a domain. (Contributed by Mario Carneiro, 15-Jun-2015.)
oppr       Domn Domn

Theoremabvn0b 16364 Another characterization of domains, hinted at in abvtriv 15931: a nonzero ring is a domain iff it has an absolute value. (Contributed by Mario Carneiro, 6-May-2015.)
AbsVal       Domn NzRing

Theoremdrngdomn 16365 A division ring is a domain. (Contributed by Mario Carneiro, 29-Mar-2015.)
Domn

Theoremisidom 16366 An integral domain is a commutative domain. (Contributed by Mario Carneiro, 17-Jun-2015.)
IDomn Domn

Theoremfldidom 16367 A field is an integral domain. (Contributed by Mario Carneiro, 29-Mar-2015.)
Field IDomn

Theoremfidomndrnglem 16368* Lemma for fidomndrng 16369. (Contributed by Mario Carneiro, 15-Jun-2015.)
r              Domn

Theoremfidomndrng 16369 A finite domain is a division ring. (Contributed by Mario Carneiro, 15-Jun-2015.)
Domn

Theoremfiidomfld 16370 A finite integral domain is a field. (Contributed by Mario Carneiro, 15-Jun-2015.)
IDomn Field

10.9  Associative algebras

10.9.1  Definition and basic properties

Syntaxcasa 16371 Associative algebra.
AssAlg

Syntaxcasp 16372 Algebraic span function.
AlgSpan

Syntaxcascl 16373 Class of algebra scalar injection function.
algSc

Definitiondf-assa 16374* 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 Scalar

Definitiondf-asp 16375* Define the algebraic span of a set of vectors in an algebra. (Contributed by Mario Carneiro, 7-Jan-2015.)
AlgSpan AssAlg SubRing

Definitiondf-ascl 16376* 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 Scalar

Theoremisassa 16377* The properties of an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.)
Scalar                            AssAlg

Theoremassalem 16378 The properties of an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.)
Scalar                            AssAlg

Theoremassaass 16379 Left-associative property of an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.)
Scalar                            AssAlg

Theoremassaassr 16380 Right-associative property of an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.)
Scalar                            AssAlg

Theoremassalmod 16381 An associative algebra is a left module. (Contributed by Mario Carneiro, 5-Dec-2014.)
AssAlg

Theoremassarng 16382 An associative algebra is a ring. (Contributed by Mario Carneiro, 5-Dec-2014.)
AssAlg

Theoremassasca 16383 An associative algebra's scalar field is a commutative ring. (Contributed by Mario Carneiro, 7-Jan-2015.)
Scalar       AssAlg

Theoremisassad 16384* Sufficient condition for being an associative algebra. (Contributed by Mario Carneiro, 5-Dec-2014.)
Scalar                                                               AssAlg

Theoremissubassa 16385 The subalgebras of an associative algebra are exactly the subrings (under the ring multiplication) that are simultaneously subspaces (under the scalar multiplication from the vector space). (Contributed by Mario Carneiro, 7-Jan-2015.)
s                             AssAlg AssAlg SubRing

Theoremsraassa 16386 The subring algebra over a commutative ring is an associative algebra. (Contributed by Mario Carneiro, 6-Oct-2015.)
subringAlg        SubRing AssAlg

Theoremrlmassa 16387 The ring module over a commutative ring is an associative algebra. (Contributed by Mario Carneiro, 6-Oct-2015.)
ringLMod AssAlg

Theoremassapropd 16388* If two structures have the same components (properties), one is an associative algebra iff the other one is. (Contributed by Mario Carneiro, 8-Feb-2015.)
Scalar       Scalar                     AssAlg AssAlg

Theoremaspval 16389* Value of the algebraic closure operation inside an associative algebra. (Contributed by Mario Carneiro, 7-Jan-2015.)
AlgSpan                     AssAlg SubRing

Theoremasplss 16390 The algebraic span of a set of vectors is a vector subspace. (Contributed by Mario Carneiro, 7-Jan-2015.)
AlgSpan                     AssAlg

Theoremaspid 16391 The algebraic span of a subalgebra is itself. (spanid 22851 analog.) (Contributed by Mario Carneiro, 7-Jan-2015.)
AlgSpan                     AssAlg SubRing

Theoremaspsubrg 16392 The algebraic span of a set of vectors is a subring of the algebra. (Contributed by Mario Carneiro, 7-Jan-2015.)
AlgSpan              AssAlg SubRing

Theoremaspss 16393 Span preserves subset ordering. (spanss 22852 analog.) (Contributed by Mario Carneiro, 7-Jan-2015.)
AlgSpan              AssAlg

Theoremaspssid 16394 A set of vectors is a subset of its span. (spanss2 22849 analog.) (Contributed by Mario Carneiro, 7-Jan-2015.)
AlgSpan              AssAlg

Theoremasclfval 16395* Function value of the algebraic scalars function. (Contributed by Mario Carneiro, 8-Mar-2015.)
algSc       Scalar

Theoremasclval 16396 Value of a mapped algebra scalar. (Contributed by Mario Carneiro, 8-Mar-2015.)
algSc       Scalar

Theoremasclfn 16397 Unconditional functionality of the algebra scalars function. (Contributed by Mario Carneiro, 9-Mar-2015.)
algSc       Scalar

Theoremasclf 16398 The algebra scalars function is a function into the base set. (Contributed by Mario Carneiro, 4-Jul-2015.)
algSc       Scalar

Theoremasclghm 16399 The algebra scalars function is a group homomorphism. (Contributed by Mario Carneiro, 4-Jul-2015.)
algSc       Scalar

Theoremasclmul1 16400 Left multiplication by a lifted scalar is the same as the scalar operation. (Contributed by Mario Carneiro, 9-Mar-2015.)
algSc       Scalar                                   AssAlg

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