Annihilators
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(5) Let R be a ring with 1 and M a left R-module. If N is a submodule of M, the annihilator of N in R is defined to be:
{r in R/rn=0 for all n in N}
Prove that the annihilator of N in R is a two-sided ideal of R.
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Solution Summary
Annihilators are investigated.
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Proof:
Let Ann(N)={r in R: rn=0 for all n in N} be the annihilator of N in R.
We want to show that Ann(N) is a two-sided ideal of R.
First, I show that Ann(N) is a subring of R. For any r,s in Ann(N), we ...
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