modern alg

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• Nov 7th 2008, 09:59 AM
wvlilgurl
modern alg
Let R be acommutative ring and let b be a fixed element of R. Prove that the set b = {r is in R and r = cb for some element c in R} is an ideal of R.
• Nov 7th 2008, 11:45 AM
ThePerfectHacker
Quote:

Originally Posted by wvlilgurl
Let R be acommutative ring and let b be a fixed element of R. Prove that the set b = {r is in R and r = cb for some element c in R} is an ideal of R.

Define $(b) = \{ rb | r\in R\}$. To show $(b)$ is an ideal of $R$ you need to show if $x,y\in (b)$ then $x\pm y \in (b)$, and also that if $r\in R$ then $r(b),(b)r \subseteq (b)$. These should be clear by definition.