Subring Generated By A Set
Let R be a ring. Any intersection of subrings of R is again a subring of R. Therefore, if X is any subset of R, the intersection of all subrings of R containing X is a subring S of R. S is the smallest subring of R containing X. ("Smallest" means that if T is any other subring of R containing X, then S is contained in T.) S is said to be the subring of R generated by X. If S = R, we may say that the ring R is generated by X.
Read more about this topic: Subring Test
Famous quotes containing the words generated and/or set:
“Here [in London, history] ... seemed the very fabric of things, as if the city were a single growth of stone and brick, uncounted strata of message and meaning, age upon age, generated over the centuries to the dictates of some now all-but-unreadable DNA of commerce and empire.”
—William Gibson (b. 1948)
“God is the immemorial refuge of the incompetent, the helpless, the miserable. They find not only sanctuary in His arms, but also a kind of superiority, soothing to their macerated egos: He will set them above their betters.”
—H.L. (Henry Lewis)