Thread: Some Radical Ideal proofs

If I is an Ideal for communitive ring R, prove Rad(I) is and Ideal of R and that I is an ideal of Rad(I)


For the first, I looked at for r ∈ R, a ∈ Rad(I). Therefore a^n ∈ I for some positive integer n.

(ra)^n = r^n*a^n ∈ I since r^n ∈ R and since I is an Ideal.

Therefore since (ra)^n ∈ I, ra ∈ Rad(I). So Rad(I) is an Ideal of R.

But for the second, I let a ∈ I, b ∈ Rad(I), therefore b^n ∈ I, but I don't know how to prove ba ∈ I.

Thanks in advance!

Originally Posted by Seda1

If I is an Ideal for communitive ring R, prove Rad(I) is and Ideal of R and that I is an ideal of Rad(I)


For the first, I looked at for r ∈ R, a ∈ Rad(I). Therefore a^n ∈ I for some positive integer n.

(ra)^n = r^n*a^n ∈ I since r^n ∈ R and since I is an Ideal.

Therefore since (ra)^n ∈ I, ra ∈ Rad(I). So Rad(I) is an Ideal of R.

But for the second, I let a ∈ I, b ∈ Rad(I), therefore b^n ∈ I, but I don't know how to prove ba ∈ I.

Thanks in advance!

Why would you have to prove ? This is always true since is an ideal...

All you need to prove is (why?)

Tonio