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

Theoremrlmmulr 16301 Ring multiplication in the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
ringLMod

Theoremrlmsca 16302 Scalars in the ring module. (Contributed by Stefan O'Rear, 6-Dec-2014.)
ScalarringLMod

Theoremrlmsca2 16303 Scalars in the ring module. (Contributed by Stefan O'Rear, 1-Apr-2015.)
ScalarringLMod

Theoremrlmvsca 16304 Scalar multiplication in the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
ringLMod

Theoremrlmtopn 16305 Topology component of the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
ringLMod

Theoremrlmds 16306 Metric component of the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
ringLMod

Theoremrlmlmod 16307 The ring module is a module. (Contributed by Stefan O'Rear, 6-Dec-2014.)
ringLMod

Theoremrlmlvec 16308 The ring module over a division ring is a vector space. (Contributed by Mario Carneiro, 4-Oct-2015.)
ringLMod

Theoremrlmvneg 16309 Vector negation in the ring module. (Contributed by Stefan O'Rear, 6-Dec-2014.) (Revised by Mario Carneiro, 5-Jun-2015.)
ringLMod

Theoremrlmscaf 16310 Functionalized scalar multiplication in the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
mulGrp ringLMod

Theoremlidlss 16311 An ideal is a subset of the base set. (Contributed by Stefan O'Rear, 28-Mar-2015.)
LIdeal

TheoremlidlssOLD 16312 An ideal is a subset of the base set. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
LIdeal

Theoremislidl 16313* Predicate of being a (left) ideal. (Contributed by Stefan O'Rear, 1-Apr-2015.)
LIdeal

Theoremlidl0cl 16314 An ideal contains 0. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LIdeal

Theoremlidlacl 16315 An ideal is closed under addition. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LIdeal

Theoremlidlnegcl 16316 An ideal contains negatives. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LIdeal

Theoremlidlsubg 16317 An ideal is a subgroup of the additive group. (Contributed by Mario Carneiro, 14-Jun-2015.)
LIdeal       SubGrp

Theoremlidlsubcl 16318 An ideal is closed under subtraction. (Contributed by Stefan O'Rear, 28-Mar-2015.)
LIdeal

Theoremlidlmcl 16319 An ideal is closed under left-multiplication by elements of the full ring. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LIdeal

Theoremlidl1el 16320 An ideal contains 1 iff it is the unit ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LIdeal

Theoremlidl0 16321 Every ring contains a zero ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LIdeal

Theoremlidl1 16322 Every ring contains a unit ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LIdeal

Theoremlidlacs 16323 The ideal system is an algebraic closure system on the base set. (Contributed by Stefan O'Rear, 4-Apr-2015.)
LIdeal       ACS

Theoremrspcl 16324 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.)
RSpan              LIdeal

Theoremrspssid 16325 The span of a set of ring elements contains those elements. (Contributed by Stefan O'Rear, 3-Jan-2015.)
RSpan

Theoremrsp1 16326 The span of the identity element is the unit ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
RSpan

Theoremrsp0 16327 The span of the zero element is the zero ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
RSpan

Theoremrspssp 16328 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.)
RSpan       LIdeal

Theoremmrcrsp 16329 Moore closure generalizes ideal span. (Contributed by Stefan O'Rear, 4-Apr-2015.)
LIdeal       RSpan       mrCls

Theoremlidlnz 16330* A nonzero ideal contains a nonzero element. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LIdeal

Theoremdrngnidl 16331 A division ring has only the two trivial ideals. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LIdeal

Theoremlidlrsppropd 16332* 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 16333 Ring two-sided ideal function.
2Ideal

Definitiondf-2idl 16334 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 16335 Definition of a two-sided ideal. (Contributed by Mario Carneiro, 14-Jun-2015.)
LIdeal       oppr       LIdeal       2Ideal

Theorem2idlcpbl 16336 The coset equivalence relation for a two-sided ideal is compatible with ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015.)
~QG        2Ideal

Theoremdivs1 16337 The multiplicative identity of the quotient ring. (Contributed by Mario Carneiro, 14-Jun-2015.)
s ~QG        2Ideal              ~QG

Theoremdivsrng 16338 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 16339* 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 16340 In a commutative ring, the left and right ideals coincide. (Contributed by Mario Carneiro, 14-Jun-2015.)
LIdeal       oppr       LIdeal

Theoremcrng2idl 16341 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 16342 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 16343 Ring left-principal-ideal function.
LPIdeal

Syntaxclpir 16344 Class of left principal ideal rings.
LPIR

Definitiondf-lpidl 16345* 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 16346 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 16347* Value of the set of principal ideals. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LPIdeal       RSpan

Theoremislpidl 16348* Property of being a principal ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LPIdeal       RSpan

Theoremlpi0 16349 The zero ideal is always principal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LPIdeal

Theoremlpi1 16350 The unit ideal is always principal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LPIdeal

Theoremislpir 16351 Principal ideal rings are where all ideals are principal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LPIdeal       LIdeal       LPIR

Theoremlpiss 16352 Principal ideals are a subclass of ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LPIdeal       LIdeal

Theoremislpir2 16353 Principal ideal rings are where all ideals are principal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LPIdeal       LIdeal       LPIR

Theoremlpirrng 16354 Principal ideal rings are rings. (Contributed by Stefan O'Rear, 24-Jan-2015.)
LPIR

Theoremdrnglpir 16355 Division rings are principal ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LPIR

Theoremrspsn 16356* 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 16357* 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 16358* 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 16359 The class of nonzero rings.
NzRing

Definitiondf-nzr 16360 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 16361 Property of a nonzero ring. (Contributed by Stefan O'Rear, 24-Feb-2015.)
NzRing

Theoremnzrnz 16362 One and zero are different in a nonzero ring. (Contributed by Stefan O'Rear, 24-Feb-2015.)
NzRing

Theoremnzrrng 16363 A nonzero ring is a ring. (Contributed by Stefan O'Rear, 24-Feb-2015.)
NzRing

Theoremdrngnzr 16364 All division rings are nonzero. (Contributed by Stefan O'Rear, 24-Feb-2015.)
NzRing

Theoremisnzr2 16365 Equivalent characterization of nonzero rings: they have at least two elements. (Contributed by Stefan O'Rear, 24-Feb-2015.)
NzRing

Theoremopprnzr 16366 The opposite of a nonzero ring is nonzero. (Contributed by Mario Carneiro, 17-Jun-2015.)
oppr       NzRing NzRing

Theoremrngelnzr 16367 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 16368 A unit is nonzero in any nonzero ring. (Contributed by Mario Carneiro, 6-Oct-2015.)
Unit              NzRing

Theoremsubrgnzr 16369 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 16370 Set of left-regular elements in a ring.
RLReg

Syntaxcdomn 16371 Class of (ring theoretic) domains.
Domn

Syntaxcidom 16372 Class of integral domains.
IDomn

Syntaxcpid 16373 Class of principal ideal domains.
PID

Definitiondf-rlreg 16374* 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 16375* 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 16376 An integral domain is a commutative domain. (Contributed by Mario Carneiro, 17-Jun-2015.)
IDomn Domn

Definitiondf-pid 16377 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 16378* Value of the set or left-regular elements in a ring. (Contributed by Stefan O'Rear, 22-Mar-2015.)
RLReg

Theoremisrrg 16379* Membership in the set of left-regular elements. (Contributed by Stefan O'Rear, 22-Mar-2015.)
RLReg

Theoremrrgeq0i 16380 Property of a left-regular element. (Contributed by Stefan O'Rear, 22-Mar-2015.)
RLReg

Theoremrrgeq0 16381 Left-multiplication by a left regular element does not change zeroness. (Contributed by Stefan O'Rear, 28-Mar-2015.)
RLReg

Theoremrrgsupp 16382 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 16383 Left-regular elements are a subset of the base set. (Contributed by Stefan O'Rear, 22-Mar-2015.)
RLReg

Theoremunitrrg 16384 Units are regular elements. (Contributed by Stefan O'Rear, 22-Mar-2015.)
RLReg       Unit

Theoremisdomn 16385* Expand definition of a domain. (Contributed by Mario Carneiro, 28-Mar-2015.)
Domn NzRing

Theoremdomnnzr 16386 A domain is a nonzero ring. (Contributed by Mario Carneiro, 28-Mar-2015.)
Domn NzRing

Theoremdomnrng 16387 A domain is a ring. (Contributed by Mario Carneiro, 28-Mar-2015.)
Domn

Theoremdomneq0 16388 In a domain, a product is zero iff it has a zero factor. (Contributed by Mario Carneiro, 28-Mar-2015.)
Domn

Theoremdomnmuln0 16389 In a domain, a product of nonzero elements is nonzero. (Contributed by Mario Carneiro, 6-May-2015.)
Domn

Theoremisdomn2 16390 A ring is a domain iff all nonzero elements are non-zero-divisors. (Contributed by Mario Carneiro, 28-Mar-2015.)
RLReg              Domn NzRing

Theoremdomnrrg 16391 In a domain, any nonzero element is a non-zero-divisor. (Contributed by Mario Carneiro, 28-Mar-2015.)
RLReg              Domn

Theoremopprdomn 16392 The opposite of a domain is also a domain. (Contributed by Mario Carneiro, 15-Jun-2015.)
oppr       Domn Domn

Theoremabvn0b 16393 Another characterization of domains, hinted at in abvtriv 15960: a nonzero ring is a domain iff it has an absolute value. (Contributed by Mario Carneiro, 6-May-2015.)
AbsVal       Domn NzRing

Theoremdrngdomn 16394 A division ring is a domain. (Contributed by Mario Carneiro, 29-Mar-2015.)
Domn

Theoremisidom 16395 An integral domain is a commutative domain. (Contributed by Mario Carneiro, 17-Jun-2015.)
IDomn Domn

Theoremfldidom 16396 A field is an integral domain. (Contributed by Mario Carneiro, 29-Mar-2015.)
Field IDomn

Theoremfidomndrnglem 16397* Lemma for fidomndrng 16398. (Contributed by Mario Carneiro, 15-Jun-2015.)
r              Domn

Theoremfidomndrng 16398 A finite domain is a division ring. (Contributed by Mario Carneiro, 15-Jun-2015.)
Domn

Theoremfiidomfld 16399 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 16400 Associative algebra.
AssAlg

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