Algebras and Rings
The definition of an associative K-algebra with unit is also frequently given in an alternative way. In this case, an algebra over a field K is a ring A together with a ring homomorphism
where Z(A) is the center of A. Since η is a ring morphism, then one must have either that A is the trivial ring, or that η is injective. This definition is equivalent to that above, with scalar multiplication
given by
Given two such associative unital K-algebras A and B, a unital K-algebra morphism f: A → B is a ring morphism that commutes with the scalar multiplication defined by η, which one may write as
for all and . In other words, the following diagram commutes:
Read more about this topic: Algebra Over A Field
Famous quotes containing the word rings:
“‘She has got rings on every finger,
Round one of them she have got three.
She have gold enough around her middle
To buy Northumberland that belongs to thee.”
—Unknown. Young Beichan (l. 61–64)