# Ring Theory/Largest two-sided ideal

Let I be a right ideal of a ring R and let A = {r in R: (R/I)r = 0}.

Prove that A is the largest two-sided ideal of R contained in I.

https://brainmass.com/math/ring-theory/ring-theory-largest-two-sided-ideal-81774

#### Solution Preview

Proof:

First, I show that A is a subring of R

For any r,s in A, we have (R/I)r=0, (R/I)s=0, then

(R/I)(r+s)=(R/I)r+(R/I)s=0+0=0, this impies r+s is in A

(R/I)rs=((R/I)r)s=0*s=0, this implies that rs is in A

So A is a subring.

Second, I show that A is a two-sided ideal.

For any r in R, s in A, we have (R/I)s=0, ...

#### Solution Summary

This solution is comprised of a detailed explanation to prove that A is the largest two-sided ideal of R contained in I.

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