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- ...\genfrac{(}{)}{}{}{a}{p}</math> as the unique nontrivial multiplicative [[homomorphism]] of <math>\mathbb{F}_p^\times</math> into <math>\mathbb{R}^\times</math>, There also exist quadratic reciprocity laws in other [[ring of integers|rings of integers]]. (I'll put that here later if I remember.)7 KB (1,182 words) - 15:46, 28 April 2016
- ...abstract algebra]], and many objects (such as all groups, as well as any [[ring]] with respect to either of its operations) are in fact monoids. ...noid <math>M</math> to the monoid <math>S^S</math>; a right operation is a homomorphism into the opposite monoid of <math>S^S</math>.3 KB (670 words) - 21:45, 21 May 2008
- ...]] of a [[ring]]. Two-sided ideals in rings are the [[kernel]]s of ring [[homomorphism]]s; in this way, two-sided ideals of rings are similar to [[normal subgroup Specifially, if <math>A</math> is a ring, a subset <math>\mathfrak{a}</math> of <math>A</math> is called a ''left id8 KB (1,389 words) - 22:44, 17 February 2020
- ...tures of the same species, for example two [[group]]s or [[field]]s. A '''homomorphism''' is a [[function]] <math>\phi : A \to B</math> that preserves the structu ...B</math> such that <math>i(a) = a</math> for all <math>a \in A</math> is a homomorphism.2 KB (303 words) - 14:33, 11 February 2024
- In general, a '''kernel''' is a measure of the failure of a [[homomorphism]] to be [[injective]]. ...] of the ring, and every two-sided ideal of a ring is the kernel of a ring homomorphism.1 KB (222 words) - 12:47, 16 June 2008
- ...ath>S</math> and <math>T</math>, respectively. An ''<math>\Omega</math>-[[homomorphism |morphism]]'' of <math>S</math> into <math>T</math> is a function <math>h: * [[Ring]]2 KB (321 words) - 08:24, 13 June 2008
- * The category '''Grp''' of all [[group|groups]], where morphisms are [[group homomorphism|group homomorphisms]]. ...Ab''' of all [[abelian group|abelian groups]], where morphisms are [[group homomorphism|group homomorphisms]].5 KB (792 words) - 18:01, 7 April 2012
- ...action as a [[homomorphism]] from <math>G</math> to <math>S_X</math>. This homomorphism is injective iff the action is faithful. ...s a set (such as the action of a group on a [[vector space]], a group, a [[ring]], a [[field]], a [[graph]], etc.). This can be done rigorously in the lang5 KB (917 words) - 20:17, 7 September 2008
- ...] <math>x</math> with coefficients in <math>R</math>." That is, it is the ring of all sums of the form The ring <math>R[x]</math> is also an <math>R</math>-module.12 KB (2,010 words) - 23:10, 2 August 2020
- A '''zero ring''' is a ring with one element, 0 (equal to 1), with the may refer to ''the'' zero ring.1 KB (221 words) - 17:44, 31 January 2022
- canonical [[homomorphism]] <math>M \to M/N</math>. Let <math>T</math> be a submodule of '''Lemma 1.''' Let <math>R</math> be a ring, and let <math>M</math> be a nonzero6 KB (1,183 words) - 14:02, 18 August 2009