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Type | Label | Description |
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Statement | ||
Theorem | iscmn 13101* | The predicate "is a commutative monoid". (Contributed by Mario Carneiro, 6-Jan-2015.) |
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Theorem | isabl2 13102* | The predicate "is an Abelian (commutative) group". (Contributed by NM, 17-Oct-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) |
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Theorem | cmnpropd 13103* | If two structures have the same group components (properties), one is a commutative monoid iff the other one is. (Contributed by Mario Carneiro, 6-Jan-2015.) |
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Theorem | ablpropd 13104* | If two structures have the same group components (properties), one is an Abelian group iff the other one is. (Contributed by NM, 6-Dec-2014.) |
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Theorem | ablprop 13105 | If two structures have the same group components (properties), one is an Abelian group iff the other one is. (Contributed by NM, 11-Oct-2013.) |
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Theorem | iscmnd 13106* | Properties that determine a commutative monoid. (Contributed by Mario Carneiro, 7-Jan-2015.) |
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Theorem | isabld 13107* | Properties that determine an Abelian group. (Contributed by NM, 6-Aug-2013.) |
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Theorem | isabli 13108* | Properties that determine an Abelian group. (Contributed by NM, 4-Sep-2011.) |
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Theorem | cmnmnd 13109 | A commutative monoid is a monoid. (Contributed by Mario Carneiro, 6-Jan-2015.) |
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Theorem | cmncom 13110 | A commutative monoid is commutative. (Contributed by Mario Carneiro, 6-Jan-2015.) |
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Theorem | ablcom 13111 | An Abelian group operation is commutative. (Contributed by NM, 26-Aug-2011.) |
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Theorem | cmn32 13112 | Commutative/associative law for commutative monoids. (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 21-Apr-2016.) |
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Theorem | cmn4 13113 | Commutative/associative law for commutative monoids. (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 21-Apr-2016.) |
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Theorem | cmn12 13114 | Commutative/associative law for commutative monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Revised by Mario Carneiro, 21-Apr-2016.) |
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Theorem | abl32 13115 | Commutative/associative law for Abelian groups. (Contributed by Stefan O'Rear, 10-Apr-2015.) (Revised by Mario Carneiro, 21-Apr-2016.) |
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Theorem | cmnmndd 13116 | A commutative monoid is a monoid. (Contributed by SN, 1-Jun-2024.) |
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Theorem | rinvmod 13117* | Uniqueness of a right inverse element in a commutative monoid, if it exists. Corresponds to caovimo 6070. (Contributed by AV, 31-Dec-2023.) |
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Theorem | ablinvadd 13118 | The inverse of an Abelian group operation. (Contributed by NM, 31-Mar-2014.) |
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Theorem | ablsub2inv 13119 | Abelian group subtraction of two inverses. (Contributed by Stefan O'Rear, 24-May-2015.) |
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Theorem | ablsubadd 13120 | Relationship between Abelian group subtraction and addition. (Contributed by NM, 31-Mar-2014.) |
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Theorem | ablsub4 13121 | Commutative/associative subtraction law for Abelian groups. (Contributed by NM, 31-Mar-2014.) |
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Theorem | abladdsub4 13122 | Abelian group addition/subtraction law. (Contributed by NM, 31-Mar-2014.) |
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Theorem | abladdsub 13123 | Associative-type law for group subtraction and addition. (Contributed by NM, 19-Apr-2014.) |
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Theorem | ablpncan2 13124 | Cancellation law for subtraction in an Abelian group. (Contributed by NM, 2-Oct-2014.) |
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Theorem | ablpncan3 13125 | A cancellation law for Abelian groups. (Contributed by NM, 23-Mar-2015.) |
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Theorem | ablsubsub 13126 | Law for double subtraction. (Contributed by NM, 7-Apr-2015.) |
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Theorem | ablsubsub4 13127 | Law for double subtraction. (Contributed by NM, 7-Apr-2015.) |
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Theorem | ablpnpcan 13128 | Cancellation law for mixed addition and subtraction. (pnpcan 8198 analog.) (Contributed by NM, 29-May-2015.) |
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Theorem | ablnncan 13129 | Cancellation law for group subtraction. (nncan 8188 analog.) (Contributed by NM, 7-Apr-2015.) |
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Theorem | ablsub32 13130 | Swap the second and third terms in a double group subtraction. (Contributed by NM, 7-Apr-2015.) |
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Theorem | ablnnncan 13131 | Cancellation law for group subtraction. (nnncan 8194 analog.) (Contributed by NM, 29-Feb-2008.) (Revised by AV, 27-Aug-2021.) |
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Theorem | ablnnncan1 13132 | Cancellation law for group subtraction. (nnncan1 8195 analog.) (Contributed by NM, 7-Apr-2015.) |
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Theorem | ablsubsub23 13133 | Swap subtrahend and result of group subtraction. (Contributed by NM, 14-Dec-2007.) (Revised by AV, 7-Oct-2021.) |
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Theorem | subcmnd 13134 | A submonoid of a commutative monoid is also commutative. (Contributed by Mario Carneiro, 10-Jan-2015.) |
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Syntax | cmgp 13135 | Multiplicative group. |
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Definition | df-mgp 13136 | Define a structure that puts the multiplication operation of a ring in the addition slot. Note that this will not actually be a group for the average ring, or even for a field, but it will be a monoid, and we get a group if we restrict to the elements that have inverses. This allows us to formalize such notions as "the multiplication operation of a ring is a monoid" or "the multiplicative identity" in terms of the identity of a monoid (df-ur 13148). (Contributed by Mario Carneiro, 21-Dec-2014.) |
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Theorem | fnmgp 13137 | The multiplicative group operator is a function. (Contributed by Mario Carneiro, 11-Mar-2015.) |
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Theorem | mgpvalg 13138 | Value of the multiplication group operation. (Contributed by Mario Carneiro, 21-Dec-2014.) |
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Theorem | mgpplusgg 13139 | Value of the group operation of the multiplication group. (Contributed by Mario Carneiro, 21-Dec-2014.) |
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Theorem | mgpex 13140 |
Existence of the multiplication group. If ![]() |
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Theorem | mgpbasg 13141 | Base set of the multiplication group. (Contributed by Mario Carneiro, 21-Dec-2014.) (Revised by Mario Carneiro, 5-Oct-2015.) |
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Theorem | mgpscag 13142 | The multiplication monoid has the same (if any) scalars as the original ring. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 5-May-2015.) |
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Theorem | mgptsetg 13143 | Topology component of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.) |
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Theorem | mgptopng 13144 | Topology of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.) |
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Theorem | mgpdsg 13145 | Distance function of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.) |
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Theorem | mgpress 13146 | Subgroup commutes with the multiplicative group operator. (Contributed by Mario Carneiro, 10-Jan-2015.) (Proof shortened by AV, 18-Oct-2024.) |
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In Wikipedia "Identity element", see https://en.wikipedia.org/wiki/Identity_element (18-Jan-2025): "... an identity with respect to multiplication is called a multiplicative identity (often denoted as 1). ... The distinction between additive and multiplicative identity is used most often for sets that support both binary operations, such as rings, integral domains, and fields. The multiplicative identity is often called unity in the latter context (a ring with unity). This should not be confused with a unit in ring theory, which is any element having a multiplicative inverse. By its own definition, unity itself is necessarily a unit." Calling the multiplicative identity of a ring a unity is taken from the definition of a ring with unity in section 17.3 of [BeauregardFraleigh] p. 135, "A ring ( R , + , . ) is a ring with unity if R is not the zero ring and ( R , . ) is a monoid. In this case, the identity element of ( R , . ) is denoted by 1 and is called the unity of R." This definition of a "ring with unity" corresponds to our definition of a unital ring (see df-ring 13186). Some authors call the multiplicative identity "unit" or "unit element" (for example in section I, 2.2 of [BourbakiAlg1] p. 14, definition in section 1.3 of [Hall] p. 4, or in section I, 1 of [Lang] p. 3), whereas other authors use the term "unit" for an element having a multiplicative inverse (for example in section 17.3 of [BeauregardFraleigh] p. 135, in definition in [Roman] p. 26, or even in section II, 1 of [Lang] p. 84). Sometimes, the multiplicative identity is simply called "one" (see, for example, chapter 8 in [Schechter] p. 180). To avoid this ambiguity of the term "unit", also mentioned in Wikipedia, we call the multiplicative identity of a structure with a multiplication (usually a ring) a "ring unity", or straightly "multiplicative identity". The term "unit" will be used for an element having a multiplicative inverse (see https://us.metamath.org/mpeuni/df-unit.html 13186 in set.mm), and we have "the ring unity is a unit", see https://us.metamath.org/mpeuni/1unit.html 13186. | ||
Syntax | cur 13147 | Extend class notation with ring unity. |
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Definition | df-ur 13148 |
Define the multiplicative identity, i.e., the monoid identity (df-0g 12712)
of the multiplicative monoid (df-mgp 13136) of a ring-like structure. This
multiplicative identity is also called "ring unity" or
"unity element".
This definition works by transferring the multiplicative operation from
the See also dfur2g 13150, which derives the "traditional" definition as the unique element of a ring which is left- and right-neutral under multiplication. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.) |
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Theorem | ringidvalg 13149 | The value of the unity element of a ring. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.) |
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Theorem | dfur2g 13150* | The multiplicative identity is the unique element of the ring that is left- and right-neutral on all elements under multiplication. (Contributed by Mario Carneiro, 10-Jan-2015.) |
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Syntax | csrg 13151 | Extend class notation with the class of all semirings. |
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Definition | df-srg 13152* | Define class of all semirings. A semiring is a set equipped with two everywhere-defined internal operations, whose first one is an additive commutative monoid structure and the second one is a multiplicative monoid structure, and where multiplication is (left- and right-) distributive over addition. Like with rings, the additive identity is an absorbing element of the multiplicative law, but in the case of semirings, this has to be part of the definition, as it cannot be deduced from distributivity alone. Definition of [Golan] p. 1. Note that our semirings are unital. Such semirings are sometimes called "rigs", being "rings without negatives". (Contributed by Thierry Arnoux, 21-Mar-2018.) |
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Theorem | issrg 13153* | The predicate "is a semiring". (Contributed by Thierry Arnoux, 21-Mar-2018.) |
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Theorem | srgcmn 13154 | A semiring is a commutative monoid. (Contributed by Thierry Arnoux, 21-Mar-2018.) |
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Theorem | srgmnd 13155 | A semiring is a monoid. (Contributed by Thierry Arnoux, 21-Mar-2018.) |
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Theorem | srgmgp 13156 | A semiring is a monoid under multiplication. (Contributed by Thierry Arnoux, 21-Mar-2018.) |
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Theorem | srgdilem 13157 | Lemma for srgdi 13162 and srgdir 13163. (Contributed by NM, 26-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
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Theorem | srgcl 13158 | Closure of the multiplication operation of a semiring. (Contributed by NM, 26-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
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Theorem | srgass 13159 | Associative law for the multiplication operation of a semiring. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
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Theorem | srgideu 13160* | The unity element of a semiring is unique. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
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Theorem | srgfcl 13161 | Functionality of the multiplication operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by AV, 24-Aug-2021.) |
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Theorem | srgdi 13162 | Distributive law for the multiplication operation of a semiring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
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Theorem | srgdir 13163 | Distributive law for the multiplication operation of a semiring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
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Theorem | srgidcl 13164 | The unity element of a semiring belongs to the base set of the semiring. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
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Theorem | srg0cl 13165 | The zero element of a semiring belongs to its base set. (Contributed by Mario Carneiro, 12-Jan-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
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Theorem | srgidmlem 13166 | Lemma for srglidm 13167 and srgridm 13168. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 27-Dec-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
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Theorem | srglidm 13167 | The unity element of a semiring is a left multiplicative identity. (Contributed by NM, 15-Sep-2011.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
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Theorem | srgridm 13168 | The unity element of a semiring is a right multiplicative identity. (Contributed by NM, 15-Sep-2011.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
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Theorem | issrgid 13169* |
Properties showing that an element ![]() |
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Theorem | srgacl 13170 | Closure of the addition operation of a semiring. (Contributed by Mario Carneiro, 14-Jan-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
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Theorem | srgcom 13171 | Commutativity of the additive group of a semiring. (Contributed by Thierry Arnoux, 1-Apr-2018.) |
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Theorem | srgrz 13172 | The zero of a semiring is a right-absorbing element. (Contributed by Thierry Arnoux, 1-Apr-2018.) |
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Theorem | srglz 13173 | The zero of a semiring is a left-absorbing element. (Contributed by AV, 23-Aug-2019.) |
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Theorem | srgisid 13174* | In a semiring, the only left-absorbing element is the additive identity. Remark in [Golan] p. 1. (Contributed by Thierry Arnoux, 1-May-2018.) |
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Theorem | srg1zr 13175 | The only semiring with a base set consisting of one element is the zero ring (at least if its operations are internal binary operations). (Contributed by FL, 13-Feb-2010.) (Revised by AV, 25-Jan-2020.) |
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Theorem | srgen1zr 13176 | The only semiring with one element is the zero ring (at least if its operations are internal binary operations). (Contributed by FL, 14-Feb-2010.) (Revised by AV, 25-Jan-2020.) |
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Theorem | srgmulgass 13177 | An associative property between group multiple and ring multiplication for semirings. (Contributed by AV, 23-Aug-2019.) |
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Theorem | srgpcomp 13178 | If two elements of a semiring commute, they also commute if one of the elements is raised to a higher power. (Contributed by AV, 23-Aug-2019.) |
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Theorem | srgpcompp 13179 | If two elements of a semiring commute, they also commute if the elements are raised to a higher power. (Contributed by AV, 23-Aug-2019.) |
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Theorem | srgpcomppsc 13180 | If two elements of a semiring commute, they also commute if the elements are raised to a higher power and a scalar multiplication is involved. (Contributed by AV, 23-Aug-2019.) |
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Theorem | srglmhm 13181* | Left-multiplication in a semiring by a fixed element of the ring is a monoid homomorphism. (Contributed by AV, 23-Aug-2019.) |
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Theorem | srgrmhm 13182* | Right-multiplication in a semiring by a fixed element of the ring is a monoid homomorphism. (Contributed by AV, 23-Aug-2019.) |
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Theorem | srg1expzeq1 13183 | The exponentiation (by a nonnegative integer) of the multiplicative identity of a semiring, analogous to mulgnn0z 13015. (Contributed by AV, 25-Nov-2019.) |
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Syntax | crg 13184 | Extend class notation with class of all (unital) rings. |
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Syntax | ccrg 13185 | Extend class notation with class of all (unital) commutative rings. |
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Definition | df-ring 13186* | Define class of all (unital) rings. A unital ring is a set equipped with two everywhere-defined internal operations, whose first one is an additive group structure and the second one is a multiplicative monoid structure, and where the addition is left- and right-distributive for the multiplication. Definition 1 in [BourbakiAlg1] p. 92 or definition of a ring with identity in part Preliminaries of [Roman] p. 19. So that the additive structure must be abelian (see ringcom 13219), care must be taken that in the case of a non-unital ring, the commutativity of addition must be postulated and cannot be proved from the other conditions. (Contributed by NM, 18-Oct-2012.) (Revised by Mario Carneiro, 27-Dec-2014.) |
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Definition | df-cring 13187 | Define class of all commutative rings. (Contributed by Mario Carneiro, 7-Jan-2015.) |
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Theorem | isring 13188* | The predicate "is a (unital) ring". Definition of "ring with unit" in [Schechter] p. 187. (Contributed by NM, 18-Oct-2012.) (Revised by Mario Carneiro, 6-Jan-2015.) |
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Theorem | ringgrp 13189 | A ring is a group. (Contributed by NM, 15-Sep-2011.) |
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Theorem | ringmgp 13190 | A ring is a monoid under multiplication. (Contributed by Mario Carneiro, 6-Jan-2015.) |
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Theorem | iscrng 13191 | A commutative ring is a ring whose multiplication is a commutative monoid. (Contributed by Mario Carneiro, 7-Jan-2015.) |
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Theorem | crngmgp 13192 | A commutative ring's multiplication operation is commutative. (Contributed by Mario Carneiro, 7-Jan-2015.) |
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Theorem | ringgrpd 13193 | A ring is a group. (Contributed by SN, 16-May-2024.) |
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Theorem | ringmnd 13194 | A ring is a monoid under addition. (Contributed by Mario Carneiro, 7-Jan-2015.) |
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Theorem | ringmgm 13195 | A ring is a magma. (Contributed by AV, 31-Jan-2020.) |
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Theorem | crngring 13196 | A commutative ring is a ring. (Contributed by Mario Carneiro, 7-Jan-2015.) |
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Theorem | crngringd 13197 | A commutative ring is a ring. (Contributed by SN, 16-May-2024.) |
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Theorem | crnggrpd 13198 | A commutative ring is a group. (Contributed by SN, 16-May-2024.) |
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Theorem | mgpf 13199 | Restricted functionality of the multiplicative group on rings. (Contributed by Mario Carneiro, 11-Mar-2015.) |
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Theorem | ringdilem 13200 | Properties of a unital ring. (Contributed by NM, 26-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) |
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