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Theorem List for Metamath Proof Explorer - 15801-15900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlidlmcl 15801 An ideal is closed under left-multiplication by elements of the full ring. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LIdeal

Theoremlidl1el 15802 An ideal contains 1 iff it is the unit ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LIdeal

Theoremlidl0 15803 Every ring contains a zero ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LIdeal

Theoremlidl1 15804 Every ring contains a unit ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LIdeal

Theoremlidlacs 15805 The ideal system is an algebraic closure system on the base set. (Contributed by Stefan O'Rear, 4-Apr-2015.)
LIdeal       ACS

Theoremrspcl 15806 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 15807 The span of a set of ring elements contains those elements. (Contributed by Stefan O'Rear, 3-Jan-2015.)
RSpan

Theoremrsp1 15808 The span of the identity element is the unit ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
RSpan

Theoremrsp0 15809 The span of the zero element is the zero ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
RSpan

Theoremrspssp 15810 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 15811 Moore closure generalizes ideal span. (Contributed by Stefan O'Rear, 4-Apr-2015.)
LIdeal       RSpan       mrCls

Theoremlidlnz 15812* A nonzero ideal contains a nonzero element. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LIdeal

Theoremdrngnidl 15813 A division ring has only the two trivial ideals. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LIdeal

Theoremlidlrsppropd 15814* 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 15815 Ring two-sided ideal function.
2Ideal

Definitiondf-2idl 15816 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 15817 Definition of a two-sided ideal. (Contributed by Mario Carneiro, 14-Jun-2015.)
LIdeal       oppr       LIdeal       2Ideal

Theorem2idlcpbl 15818 The coset equivalence relation for a two-sided ideal is compatible with ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015.)
~QG        2Ideal

Theoremdivs1 15819 The multiplicative identity of the quotient ring. (Contributed by Mario Carneiro, 14-Jun-2015.)
s ~QG        2Ideal              ~QG

Theoremdivsrng 15820 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 15821* 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 15822 In a commutative ring, the left and right ideals coincide. (Contributed by Mario Carneiro, 14-Jun-2015.)
LIdeal       oppr       LIdeal

Theoremcrng2idl 15823 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 15824 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 15825 Ring left-principal-ideal function.
LPIdeal

Syntaxclpir 15826 Class of left principal ideal rings.
LPIR

Definitiondf-lpidl 15827* 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 15828 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 15829* Value of the set of principal ideals. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LPIdeal       RSpan

Theoremislpidl 15830* Property of being a principal ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LPIdeal       RSpan

Theoremlpi0 15831 The zero ideal is always principal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LPIdeal

Theoremlpi1 15832 The unit ideal is always principal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LPIdeal

Theoremislpir 15833 Principal ideal rings are where all ideals are principal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LPIdeal       LIdeal       LPIR

Theoremlpiss 15834 Principal ideals are a subclass of ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LPIdeal       LIdeal

Theoremislpir2 15835 Principal ideal rings are where all ideals are principal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LPIdeal       LIdeal       LPIR

Theoremlpirrng 15836 Principal ideal rings are rings. (Contributed by Stefan O'Rear, 24-Jan-2015.)
LPIR

Theoremdrnglpir 15837 Division rings are principal ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LPIR

Theoremrspsn 15838* 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 15839* 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 15840* 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 15841 The class of nonzero rings.
NzRing

Definitiondf-nzr 15842 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 15843 Property of a nonzero ring. (Contributed by Stefan O'Rear, 24-Feb-2015.)
NzRing

Theoremnzrnz 15844 One and zero are different in a nonzero ring. (Contributed by Stefan O'Rear, 24-Feb-2015.)
NzRing

Theoremnzrrng 15845 A nonzero ring is a ring. (Contributed by Stefan O'Rear, 24-Feb-2015.)
NzRing

Theoremdrngnzr 15846 All division rings are nonzero. (Contributed by Stefan O'Rear, 24-Feb-2015.)
NzRing

Theoremisnzr2 15847 Equivalent characterization of nonzero rings: they have at least two elements. (Contributed by Stefan O'Rear, 24-Feb-2015.)
NzRing

Theoremopprnzr 15848 The opposite of a nonzero ring is nonzero. (Contributed by Mario Carneiro, 17-Jun-2015.)
oppr       NzRing NzRing

Theoremrngelnzr 15849 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 15850 A unit is nonzero in any nonzero ring. (Contributed by Mario Carneiro, 6-Oct-2015.)
Unit              NzRing

Theoremsubrgnzr 15851 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 ring

Syntaxcrlreg 15852 Set of left-regular elements in a ring.
RLReg

Syntaxcdomn 15853 Class of (ring theoretic) domains.
Domn

Syntaxcidom 15854 Class of integral domains.
IDomn

Syntaxcpid 15855 Class of principal ideal domains.
PID

Definitiondf-rlreg 15856* 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 15857* 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 15858 An integral domain is a commutative domain. (Contributed by Mario Carneiro, 17-Jun-2015.)
IDomn Domn

Definitiondf-pid 15859 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 15860* Value of the set or left-regular elements in a ring. (Contributed by Stefan O'Rear, 22-Mar-2015.)
RLReg

Theoremisrrg 15861* Membership in the set of left-regular elements. (Contributed by Stefan O'Rear, 22-Mar-2015.)
RLReg

Theoremrrgeq0i 15862 Property of a left-regular element. (Contributed by Stefan O'Rear, 22-Mar-2015.)
RLReg

Theoremrrgeq0 15863 Left-multiplication by a left regular element does not change zeroness. (Contributed by Stefan O'Rear, 28-Mar-2015.)
RLReg

Theoremrrgsupp 15864 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 15865 Left-regular elements are a subset of the base set. (Contributed by Stefan O'Rear, 22-Mar-2015.)
RLReg

Theoremunitrrg 15866 Units are regular elements. (Contributed by Stefan O'Rear, 22-Mar-2015.)
RLReg       Unit

Theoremisdomn 15867* Expand definition of a domain. (Contributed by Mario Carneiro, 28-Mar-2015.)
Domn NzRing

Theoremdomnnzr 15868 A domain is a nonzero ring. (Contributed by Mario Carneiro, 28-Mar-2015.)
Domn NzRing

Theoremdomnrng 15869 A domain is a ring. (Contributed by Mario Carneiro, 28-Mar-2015.)
Domn

Theoremdomneq0 15870 In a domain, a product is zero iff it has a zero factor. (Contributed by Mario Carneiro, 28-Mar-2015.)
Domn

Theoremdomnmuln0 15871 In a domain, a product of nonzero elements is nonzero. (Contributed by Mario Carneiro, 6-May-2015.)
Domn

Theoremisdomn2 15872 A ring is a domain iff all nonzero elements are non-zero-divisors. (Contributed by Mario Carneiro, 28-Mar-2015.)
RLReg              Domn NzRing

Theoremdomnrrg 15873 In a domain, any nonzero element is a non-zero-divisor. (Contributed by Mario Carneiro, 28-Mar-2015.)
RLReg              Domn

Theoremopprdomn 15874 The opposite of a domain is also a domain. (Contributed by Mario Carneiro, 15-Jun-2015.)
oppr       Domn Domn

Theoremabvn0b 15875 Another characterization of domains, hinted at in abvtriv 15441: a nonzero ring is a domain iff it has an absolute value. (Contributed by Mario Carneiro, 6-May-2015.)
AbsVal       Domn NzRing

Theoremdrngdomn 15876 A division ring is a domain. (Contributed by Mario Carneiro, 29-Mar-2015.)
Domn

Theoremisidom 15877 An integral domain is a commutative domain. (Contributed by Mario Carneiro, 17-Jun-2015.)
IDomn Domn

Theoremfldidom 15878 A field is an integral domain. (Contributed by Mario Carneiro, 29-Mar-2015.)
Field IDomn

Theoremfidomndrnglem 15879* Lemma for fidomndrng 15880. (Contributed by Mario Carneiro, 15-Jun-2015.)
r              Domn

Theoremfidomndrng 15880 A finite domain is a division ring. (Contributed by Mario Carneiro, 15-Jun-2015.)
Domn

Theoremfiidomfld 15881 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 15882 Associative algebra.
AssAlg

Syntaxcasp 15883 Algebraic span function.
AlgSpan

Syntaxcascl 15884 Class of algebra scalar injection function.
algSc

Definitiondf-assa 15885* Definition of an associative algebra. An associative algebra is a set equipped with a left-module structure on a (commutative) ring, coupled with an 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 15886* Define the algebraic span of a set of vectors in an algebra. (Contributed by Mario Carneiro, 7-Jan-2015.)
AlgSpan AssAlg SubRing

Definitiondf-ascl 15887* 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 15888* The properties of an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.)
Scalar                            AssAlg

Theoremassalem 15889 The properties of an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.)
Scalar                            AssAlg

Theoremassaass 15890 Left-associative property of an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.)
Scalar                            AssAlg

Theoremassaassr 15891 Right-associative property of an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.)
Scalar                            AssAlg

Theoremassalmod 15892 An associative algebra is a left module. (Contributed by Mario Carneiro, 5-Dec-2014.)
AssAlg

Theoremassarng 15893 An associative algebra is a ring. (Contributed by Mario Carneiro, 5-Dec-2014.)
AssAlg

Theoremassasca 15894 An associative algebra's scalar field is a commutative ring. (Contributed by Mario Carneiro, 7-Jan-2015.)
Scalar       AssAlg

Theoremisassad 15895* Sufficient condition for being an associative algebra. (Contributed by Mario Carneiro, 5-Dec-2014.)
Scalar                                                               AssAlg

Theoremissubassa 15896 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 15897 The subring algebra over a commutative ring is an associative algebra. (Contributed by Mario Carneiro, 6-Oct-2015.)
subringAlg        SubRing AssAlg

Theoremrlmassa 15898 The ring module over a commutative ring is an associative algebra. (Contributed by Mario Carneiro, 6-Oct-2015.)
ringLMod AssAlg

Theoremassapropd 15899* 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 15900* Value of the algebraic closure operation inside an associative algebra. (Contributed by Mario Carneiro, 7-Jan-2015.)
AlgSpan                     AssAlg SubRing

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