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## Homework Statement

a) If A is an ideal of R and B is an ideal of S.

Show that A x B is an ideal of R x S.

b) Show that every ideal C of RxS has the form C = AxB as in(a)

[hint: A = { a in R | (a,0) in C}]

## The Attempt at a Solution

a)Since A and B are ideal of R and S, aR and Ra are subsets of A, bS and Sb are subsets of B.

Let (a,b) in AxB and (r,s) in RxS, (a,b)(r,s) = (ar,bs) in AxB since ar in A and bs in B.

b) Let A = { a in R | (a,0) in C} and

B = { b in S | (0,b) in C}

We need to show that AxB = <(a,0),(0,b)>.

my idea is correct?