Relation To Ideals
Proper ideals are subrings that are closed under both left and right multiplication by elements from R.
If one omits the requirement that rings have a unity element, then subrings need only be non-empty and otherwise conform to the ring structure, and ideals become subrings. Ideals may or may not have their own multiplicative identity (distinct from the identity of the ring):
- The ideal I = {(z,0) | z in Z} of the ring Z × Z = {(x,y) | x,y in Z} with componentwise addition and multiplication has the identity (1,0), which is different from the identity (1,1) of the ring. So I is a ring with unity, and a "subring-without-unity", but not a "subring-with-unity" of Z × Z.
- The proper ideals of Z have no multiplicative identity.
Read more about this topic: Subring Test
Famous quotes containing the words relation to, relation and/or ideals:
“Whoever has a keen eye for profits, is blind in relation to his craft.”
—Sophocles (497–406/5 B.C.)
“You know there are no secrets in America. It’s quite different in England, where people think of a secret as a shared relation between two people.”
—W.H. (Wystan Hugh)
“But I would emphasize again that social and economic solutions, as such, will not avail to satisfy the aspirations of the people unless they conform with the traditions of our race, deeply grooved in their sentiments through a century and a half of struggle for ideals of life that are rooted in religion and fed from purely spiritual springs.”
—Herbert Hoover (1874–1964)