Ring Homomorphisms
As usual in algebra, a function f between two objects that respects the structures of the objects in question is called homomorphism. In the case of rings, a ring homomorphism is a map f : R → S such that
- f(a + b) = f(a) + f(b), f(ab) = f(a)f(b) and f(1) = 1.
These conditions ensure f(0) = 0, but the requirement that the multiplicative identity element 1 is preserved under f would not follow from the two remaining properties. In such a situation S is also called an R-algebra, by understanding that s in S may be multiplied by some r of R, by setting
- r · s := f(r) · s.
The kernel and image of f are defined by ker (f) = {r ∈ R, f(r) = 0} and im (f) = f(R) = {f(r), r ∈ R}. The kernel is an ideal of R, and the image is a subring of S.
Read more about this topic: Commutative Ring
Famous quotes containing the word ring:
“He will not idly dance at his work who has wood to cut and cord before nightfall in the short days of winter; but every stroke will be husbanded, and ring soberly through the wood; and so will the strokes of that scholar’s pen, which at evening record the story of the day, ring soberly, yet cheerily, on the ear of the reader, long after the echoes of his axe have died away.”
—Henry David Thoreau (1817–1862)