Answer to Question #308642 in Abstract Algebra for Kingwales

Question

Prove that an ideal M neq R in a commutative ring R with identity is maximal if and only if for every r in R-M, there exists x in R such that 1_R - rx in M.

Answer 1

Let R be a commutative ring with identity. An R module M is said to be a multiplication module if every Sub module of M is of the form IM

For some ideal I of R, we shall call a ring R a regular multiplication ring is a ring in which every ideal is a Multiplcation Module....it is well known that Multiplication domains are precisely dedekind domains

Proof: let I be a regular ideal of R then I = P1 P2......Pn where P1 P2 ...Pn are distinct maximal ideals of R and V1, V2...Vn are positive integers

Hence R/I= (+); R/P1 since for each i=

1,2 n P1 is a regular ideal of R