- #1
hsong9
- 80
- 1
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?