Hello, thank you very much for reading.

I have two homework assignments which I struggle with for quite some time.

1) Let R be acommutative, unitary(R has a unit) Ring,and let I be an Ideal to R.

I was given the definition of theof I:radical

Rad(I) = {x in R| there exists n in N such that x^n is in I}

I needed to prove that Rad(I) is an ideal in R, that Rad(Rad(I))=Rad(I), and that the set of all nilpotent elements N(R) (such elements x that have n in N such that x^n is 0) is an ideal.

So I've proved all of those. Rad(I) is an ideal by definition, and it took some time. Rad(Rad(I))=Rad(I) was easy. The third claim is also easy, noticing that this particular set N(R) is actually Rad({0}), which I've just proved is an ideal (since {0} is an ideal).

Now - the last question is: assuming that R isnotcommutative - which of the claims is correct?

I'm sure that Rad(Rad(I))=Rad(I) either way - I didn't use commutativness here.

However, I can't see if either the first or the third claims are correct here. First of all, if the first is still correct, the third would also be.

That's why I tend to think the first isn't correct, and the third is. However, I didn't succeed in proving the third (that N(R) is an ideal even though R isn't commutative), and I can't find a counter-example for the first claim (that Rad(I) isnotan ideal).

So that's one question

2) (much shorter!)

I'm given a ring R, a sub-ring S of R, and an ideal I of R.

I proved that the intersection of I and S is an ideal in S.

Now they ask:

Is every ideal in S an intersection of S with an ideal in R?

Cannot show either it's correct or false :-\

It reminds me of the definition of a closure of a set in a subset in topology, but unfortunately I make any analogies here...

THANK YOU VERY MUCH FOR READING!!!

Bless you all

Love,

Tomer.