criterion for cyclic rings to be principal ideal rings
Let be a cyclic ring. If has a multiplicative identity , then generates (http://planetmath.org/Generator) the additive group of . Let be an ideal of . Since is principal, it may be assumed that contains a nonzero element. Let be the smallest natural number such that . The inclusion is trivial. Let . Since , there exists with . By the division algorithm, there exists with such that . Thus, . Since , by choice of , it must be the case that . Thus, . Hence, , and is a principal ideal ring.
Conversely, if is a principal ideal ring, then is a principal ideal. Let be the behavior of and be a generator (http://planetmath.org/Generator) of the additive group of such that . Since is principal, there exists such that . Let such that . Since , there exists with . Let such that . Then . If is infinite, then , in which case since is nonnegative. If is finite, then , in which case since is a positive divisor of . In either case, has behavior one, and it follows that has a multiplicative identity. ∎
|Title||criterion for cyclic rings to be principal ideal rings|
|Date of creation||2013-03-22 15:57:03|
|Last modified on||2013-03-22 15:57:03|
|Last modified by||Wkbj79 (1863)|