- #1
wnorman27
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Homework Statement
If a ring R contains two ideals B and C with B+C=R and B[itex]\cap[/itex]C=0, prove that B and C are rings and R[itex]\cong[/itex]B x C.
Homework Equations
B+C={all b+c|b[itex]\in[/itex]B and c[itex]\in[/itex]C}
The Attempt at a Solution
So far I've discovered that if the unit of R is in one of the ideals then that ideal is all of R, while the other is just {0}. Unfortunately this doesn't help much because I can't guarantee the unit of R is in either ideal. In fact I'm having trouble showing that anything is in B or C. I've been trying to find units for B and C (distinct from the unit of R since while B and C are ring,s they are not necessarily subrings). Is this a faulty approach? I can't think of any other approach...