Homomorphism of commutative unital rings: Difference between revisions

Latest revision as of 16:23, 12 May 2008

Definition

Let $R,S$ be commutative unital rings. A function $f:R\to S$ is termed a homomorphism of commutative unital rings, or simply a homomorphism, if it satisfies the following conditions:

$f(a+b)=f(a)+f(b)$
$f(0)=0$
$f(-a)--f(a)$
$f(1)=1$
$f(ab)=f(a)f(b)$

It turns out that conditions (2) and (3) follow from (1). However, condition (4) does not follow from condition (5). One comes across situations where a map of commutative unital rings preserves the additive and multiplicative structure but does not send the multiplicative identity to the multiplicative identity; such a map is not a homomorphism of commutative unital rings.

Revision as of 00:06, 3 February 2008 (view source) Vipul (talk \| contribs) (New page: ==Definition== Let <math>R,S</math> be commutative unital rings. A function <math>f:R \to S</math> is termed a '''homomorphism of commutative unital rings''', or simply a '''homomorph...)	Latest revision as of 16:23, 12 May 2008 (view source) Vipul (talk \| contribs) m (1 revision)
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