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Type | Label | Description |
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Statement | ||
Theorem | rngimrcl 40001 | Reverse closure for an isomorphism of non-unital rings. (Contributed by AV, 22-Feb-2020.) |
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Theorem | rnghmghm 40002 | A non-unital ring homomorphism is an additive group homomorphism. (Contributed by AV, 23-Feb-2020.) |
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Theorem | rnghmf 40003 | A ring homomorphism is a function. (Contributed by AV, 23-Feb-2020.) |
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Theorem | rnghmmul 40004 | A homomorphism of non-unital rings preserves multiplication. (Contributed by AV, 23-Feb-2020.) |
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Theorem | isrnghm2d 40005* | Demonstration of non-unital ring homomorphism. (Contributed by AV, 23-Feb-2020.) |
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Theorem | isrnghmd 40006* | Demonstration of non-unital ring homomorphism. (Contributed by AV, 23-Feb-2020.) |
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Theorem | rnghmf1o 40007 | A non-unital ring homomorphism is bijective iff its converse is also a non-unital ring homomorphism. (Contributed by AV, 27-Feb-2020.) |
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Theorem | isrngim 40008 | An isomorphism of non-unital rings is a bijective homomorphism. (Contributed by AV, 23-Feb-2020.) |
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Theorem | rngimf1o 40009 | An isomorphism of non-unital rings is a bijection. (Contributed by AV, 23-Feb-2020.) |
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Theorem | rngimrnghm 40010 | An isomorphism of non-unital rings is a homomorphism. (Contributed by AV, 23-Feb-2020.) |
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Theorem | rnghmco 40011 | The composition of non-unital ring homomorphisms is a homomorphism. (Contributed by AV, 27-Feb-2020.) |
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Theorem | idrnghm 40012 | The identity homomorphism on a non-unital ring. (Contributed by AV, 27-Feb-2020.) |
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Theorem | c0mgm 40013* | The constant mapping to zero is a magma homomorphism into a monoid. Remark: Instead of the assumption that T is a monoid, it would be sufficient that T is a magma with a right or left identity. (Contributed by AV, 17-Apr-2020.) |
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Theorem | c0mhm 40014* | The constant mapping to zero is a monoid homomorphism. (Contributed by AV, 16-Apr-2020.) |
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Theorem | c0ghm 40015* | The constant mapping to zero is a group homomorphism. (Contributed by AV, 16-Apr-2020.) |
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Theorem | c0rhm 40016* | The constant mapping to zero is a ring homomorphism from any ring to the zero ring. (Contributed by AV, 17-Apr-2020.) |
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Theorem | c0rnghm 40017* | The constant mapping to zero is a nonunital ring homomorphism from any nonunital ring to the zero ring. (Contributed by AV, 17-Apr-2020.) |
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Theorem | c0snmgmhm 40018* | The constant mapping to zero is a magma homomorphism from a magma with one element to any monoid. (Contributed by AV, 17-Apr-2020.) |
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Theorem | c0snmhm 40019* | The constant mapping to zero is a monoid homomorphism from the trivial monoid (consisting of the zero only) to any monoid. (Contributed by AV, 17-Apr-2020.) |
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Theorem | c0snghm 40020* | The constant mapping to zero is a group homomorphism from the trivial group (consisting of the zero only) to any group. (Contributed by AV, 17-Apr-2020.) |
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Theorem | zrrnghm 40021* | The constant mapping to zero is a nonunital ring homomorphism from the zero ring to any nonunital ring. (Contributed by AV, 17-Apr-2020.) |
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Theorem | rhmfn 40022 | The mapping of two rings to the ring homomorphisms between them is a function. (Contributed by AV, 1-Mar-2020.) |
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Theorem | rhmval 40023 | The ring homomorphisms between two rings. (Contributed by AV, 1-Mar-2020.) |
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Theorem | rhmisrnghm 40024 | Each unital ring homomorphism is a non-unital ring homomorphism. (Contributed by AV, 29-Feb-2020.) |
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Theorem | lidldomn1 40025* | If a (left) ideal (which is not the zero ideal) of a domain has a multiplicative identity element, the identity element is the identity of the domain. (Contributed by AV, 17-Feb-2020.) |
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Theorem | lidlssbas 40026 | The base set of the restriction of the ring to a (left) ideal is a subset of the base set of the ring. (Contributed by AV, 17-Feb-2020.) |
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Theorem | lidlbas 40027 | A (left) ideal of a ring is the base set of the restriction of the ring to this ideal. (Contributed by AV, 17-Feb-2020.) |
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Theorem | lidlabl 40028 | A (left) ideal of a ring is an (additive) abelian group. (Contributed by AV, 17-Feb-2020.) |
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Theorem | lidlmmgm 40029 | The multiplicative group of a (left) ideal of a ring is a magma. (Contributed by AV, 17-Feb-2020.) |
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Theorem | lidlmsgrp 40030 | The multiplicative group of a (left) ideal of a ring is a semigroup. (Contributed by AV, 17-Feb-2020.) |
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Theorem | lidlrng 40031 | A (left) ideal of a ring is a non-unital ring. (Contributed by AV, 17-Feb-2020.) |
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Theorem | zlidlring 40032 | The zero (left) ideal of a non-unital ring is a unital ring (the zero ring). (Contributed by AV, 16-Feb-2020.) |
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Theorem | uzlidlring 40033 | Only the zero (left) ideal or the unit (left) ideal of a domain is a unital ring. (Contributed by AV, 18-Feb-2020.) |
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Theorem | lidldomnnring 40034 | A (left) ideal of a domain which is neither the zero ideal nor the unit ideal is not a unital ring. (Contributed by AV, 18-Feb-2020.) |
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Theorem | 0even 40035* | 0 is an even integer. (Contributed by AV, 11-Feb-2020.) |
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Theorem | 1neven 40036* | 1 is not an even integer. (Contributed by AV, 12-Feb-2020.) |
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Theorem | 2even 40037* | 2 is an even integer. (Contributed by AV, 12-Feb-2020.) |
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Theorem | 2zlidl 40038* | The even integers are a (left) ideal of the ring of integers. (Contributed by AV, 20-Feb-2020.) |
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Theorem | 2zrng 40039* | The ring of integers restricted to the even integers is a (non-unital) ring, the "ring of even integers". Remark: the structure of the complementary subset of the set of integers, the odd integers, is not even a magma, see oddinmgm 39919. (Contributed by AV, 20-Feb-2020.) |
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Theorem | 2zrngbas 40040* | The base set of R is the set of all even integers. (Contributed by AV, 31-Jan-2020.) |
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Theorem | 2zrngadd 40041* | The group addition operation of R is the addition of complex numbers. (Contributed by AV, 31-Jan-2020.) |
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Theorem | 2zrng0 40042* | The additive identity of R is the complex number 0. (Contributed by AV, 11-Feb-2020.) |
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Theorem | 2zrngamgm 40043* | R is an (additive) magma. (Contributed by AV, 6-Jan-2020.) |
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Theorem | 2zrngasgrp 40044* | R is an (additive) semigroup. (Contributed by AV, 4-Feb-2020.) |
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Theorem | 2zrngamnd 40045* | R is an (additive) monoid. (Contributed by AV, 11-Feb-2020.) |
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Theorem | 2zrngacmnd 40046* | R is a commutative (additive) monoid. (Contributed by AV, 11-Feb-2020.) |
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Theorem | 2zrngagrp 40047* | R is an (additive) group. (Contributed by AV, 6-Jan-2020.) |
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Theorem | 2zrngaabl 40048* | R is an (additive) abelian group. (Contributed by AV, 11-Feb-2020.) |
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Theorem | 2zrngmul 40049* | The ring multiplication operation of R is the multiplication on complex numbers. (Contributed by AV, 31-Jan-2020.) |
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Theorem | 2zrngmmgm 40050* | R is a (multiplicative) magma. (Contributed by AV, 11-Feb-2020.) |
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Theorem | 2zrngmsgrp 40051* | R is a (multiplicative) semigroup. (Contributed by AV, 4-Feb-2020.) |
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Theorem | 2zrngALT 40052* | The ring of integers restricted to the even integers is a (non-unital) ring, the "ring of even integers". Alternate version of 2zrng 40039, based on a restriction of the field of the complex numbers. The proof is based on the facts that the ring of even integers is an additive abelian group (see 2zrngaabl 40048) and a multiplicative semigroup (see 2zrngmsgrp 40051). (Contributed by AV, 11-Feb-2020.) (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | 2zrngnmlid 40053* | R has no multiplicative (left) identity. (Contributed by AV, 12-Feb-2020.) |
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Theorem | 2zrngnmrid 40054* | R has no multiplicative (right) identity. (Contributed by AV, 12-Feb-2020.) |
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Theorem | 2zrngnmlid2 40055* | R has no multiplicative (left) identity. (Contributed by AV, 12-Feb-2020.) |
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Theorem | 2zrngnring 40056* | R is not a unital ring. (Contributed by AV, 6-Jan-2020.) |
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Theorem | plusgndxnmulrndx 40057 | The slot for the group (addition) operation is not the slot for the ring (multiplication) operation in an extensible structure. (Contributed by AV, 16-Feb-2020.) |
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Theorem | basendxnmulrndx 40058 | The slot for the base set is not the slot for the ring (multiplication) operation in an extensible structure. (Contributed by AV, 16-Feb-2020.) |
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Theorem | cznrnglem 40059 | Lemma for cznrng 40061: The base set of the ring constructed from a ℤ/nℤ structure by replacing the (multiplicative) ring operation by a constant operation is the base set of the ℤ/nℤ structure. (Contributed by AV, 16-Feb-2020.) |
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Theorem | cznabel 40060 | The ring constructed from a ℤ/nℤ structure by replacing the (multiplicative) ring operation by a constant operation is an abelian group. (Contributed by AV, 16-Feb-2020.) |
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Theorem | cznrng 40061* | The ring constructed from a ℤ/nℤ structure by replacing the (multiplicative) ring operation by a constant operation is a non-unital ring. (Contributed by AV, 17-Feb-2020.) |
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Theorem | cznnring 40062* |
The ring constructed from a ℤ/nℤ structure with ![]() ![]() ![]() |
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The "category of non-unital rings" RngCat is the category of all non-unital rings Rng in a universe and non-unital ring homomorphisms RngHomo between these rings. This category is defined as "category restriction" of the category of extensible structures ExtStrCat, which restricts the objects to non-unital rings and the morphisms to the non-unital ring homomorphisms, while the composition of morphisms is preserved, see df-rngc 40065. Alternatively, the category of non-unital rings could have been defined as extensible structure consisting of three components/slots for the objects, morphisms and composition, see df-rngcALTV 40066 or dfrngc2 40078.
Since we consider only "small categories" (i.e. categories whose
objects and
morphisms are actually sets and not proper classes), the objects of the
category (i.e. the base set of the category regarded as extensible structure)
are a subset of the non-unital rings (relativized to a subset or
"universe"
By showing that the non-unital ring homomorphisms between non-unital rings are
a subcategory subset ( | ||
Syntax | crngc 40063 | Extend class notation to include the category Rng. |
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Syntax | crngcALTV 40064 | Extend class notation to include the category Rng. (New usage is discouraged.) |
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Definition | df-rngc 40065 |
Definition of the category Rng, relativized to a subset ![]() ![]() ![]() |
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Definition | df-rngcALTV 40066* |
Definition of the category Rng, relativized to a subset ![]() ![]() ![]() |
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Theorem | rngcvalALTV 40067* | Value of the category of non-unital rings (in a universe). (New usage is discouraged.) (Contributed by AV, 27-Feb-2020.) |
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Theorem | rngcval 40068 | Value of the category of non-unital rings (in a universe). (Contributed by AV, 27-Feb-2020.) (Revised by AV, 8-Mar-2020.) |
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Theorem | rnghmresfn 40069 | The class of non-unital ring homomorphisms restricted to subsets of non-unital rings is a function. (Contributed by AV, 4-Mar-2020.) |
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Theorem | rnghmresel 40070 | An element of the non-unital ring homomorphisms restricted to a subset of non-unital rings is a non-unital ring homomorphisms. (Contributed by AV, 9-Mar-2020.) |
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Theorem | rngcbas 40071 | Set of objects of the category of non-unital rings (in a universe). (Contributed by AV, 27-Feb-2020.) (Revised by AV, 8-Mar-2020.) |
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Theorem | rngchomfval 40072 | Set of arrows of the category of non-unital rings (in a universe). (Contributed by AV, 27-Feb-2020.) (Revised by AV, 8-Mar-2020.) |
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Theorem | rngchom 40073 | Set of arrows of the category of non-unital rings (in a universe). (Contributed by AV, 27-Feb-2020.) |
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Theorem | elrngchom 40074 | A morphism of non-unital rings is a function. (Contributed by AV, 27-Feb-2020.) |
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Theorem | rngchomfeqhom 40075 | The functionalized Hom-set operation equals the Hom-set operation in the category of non-unital rings (in a universe). (Contributed by AV, 9-Mar-2020.) |
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Theorem | rngccofval 40076 | Composition in the category of non-unital rings. (Contributed by AV, 27-Feb-2020.) (Revised by AV, 8-Mar-2020.) |
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Theorem | rngcco 40077 | Composition in the category of non-unital rings. (Contributed by AV, 27-Feb-2020.) (Revised by AV, 8-Mar-2020.) |
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Theorem | dfrngc2 40078 | Alternate definition of the category of non-unital rings (in a universe). (Contributed by AV, 16-Mar-2020.) |
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Theorem | rnghmsscmap2 40079* | The non-unital ring homomorphisms between non-unital rings (in a universe) are a subcategory subset of the mappings between base sets of non-unital rings (in the same universe). (Contributed by AV, 6-Mar-2020.) |
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Theorem | rnghmsscmap 40080* | The non-unital ring homomorphisms between non-unital rings (in a universe) are a subcategory subset of the mappings between base sets of extensible structures (in the same universe). (Contributed by AV, 9-Mar-2020.) |
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Theorem | rnghmsubcsetclem1 40081 | Lemma 1 for rnghmsubcsetc 40083. (Contributed by AV, 9-Mar-2020.) |
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Theorem | rnghmsubcsetclem2 40082* | Lemma 2 for rnghmsubcsetc 40083. (Contributed by AV, 9-Mar-2020.) |
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Theorem | rnghmsubcsetc 40083 | The non-unital ring homomorphisms between non-unital rings (in a universe) are a subcategory of the category of extensible structures. (Contributed by AV, 9-Mar-2020.) |
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Theorem | rngccat 40084 | The category of non-unital rings is a category. (Contributed by AV, 27-Feb-2020.) (Revised by AV, 9-Mar-2020.) |
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Theorem | rngcid 40085 | The identity arrow in the category of non-unital rings is the identity function. (Contributed by AV, 27-Feb-2020.) (Revised by AV, 10-Mar-2020.) |
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Theorem | rngcsect 40086 | A section in the category of non-unital rings, written out. (Contributed by AV, 28-Feb-2020.) |
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Theorem | rngcinv 40087 | An inverse in the category of non-unital rings is the converse operation. (Contributed by AV, 28-Feb-2020.) |
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Theorem | rngciso 40088 | An isomorphism in the category of non-unital rings is a bijection. (Contributed by AV, 28-Feb-2020.) |
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Theorem | rngcbasALTV 40089 | Set of objects of the category of non-unital rings (in a universe). (New usage is discouraged.) (Contributed by AV, 27-Feb-2020.) |
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Theorem | rngchomfvalALTV 40090* | Set of arrows of the category of non-unital rings (in a universe). (New usage is discouraged.) (Contributed by AV, 27-Feb-2020.) |
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Theorem | rngchomALTV 40091 | Set of arrows of the category of non-unital rings (in a universe). (New usage is discouraged.) (Contributed by AV, 27-Feb-2020.) |
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Theorem | elrngchomALTV 40092 | A morphism of non-unital rings is a function. (New usage is discouraged.) (Contributed by AV, 27-Feb-2020.) |
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Theorem | rngccofvalALTV 40093* | Composition in the category of non-unital rings. (New usage is discouraged.) (Contributed by AV, 27-Feb-2020.) |
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Theorem | rngccoALTV 40094 | Composition in the category of non-unital rings. (New usage is discouraged.) (Contributed by AV, 27-Feb-2020.) |
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Theorem | rngccatidALTV 40095* | Lemma for rngccatALTV 40096. (New usage is discouraged.) (Contributed by AV, 27-Feb-2020.) |
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Theorem | rngccatALTV 40096 | The category of non-unital rings is a category. (Contributed by AV, 27-Feb-2020.) (New usage is discouraged.) |
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Theorem | rngcidALTV 40097 | The identity arrow in the category of non-unital rings is the identity function. (Contributed by AV, 27-Feb-2020.) (New usage is discouraged.) |
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Theorem | rngcsectALTV 40098 | A section in the category of non-unital rings, written out. (Contributed by AV, 28-Feb-2020.) (New usage is discouraged.) |
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Theorem | rngcinvALTV 40099 | An inverse in the category of non-unital rings is the converse operation. (Contributed by AV, 28-Feb-2020.) (New usage is discouraged.) |
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Theorem | rngcisoALTV 40100 | An isomorphism in the category of non-unital rings is a bijection. (Contributed by AV, 28-Feb-2020.) (New usage is discouraged.) |
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