Operations on Ideals - Hello Experts

[DukeSteve]

Hello Experts,

I post this question here because in the homework topics there is no abstract algebra!
Please help me I want to understand it:

I have a ring R with unit. Also I am given n - natural number, I_n is the set {x in R: n*x = 0}

I have to prove or refute: Given n, m natural numbers:

A) Is I_n + I_m is an ideal of the form of I_k?
B) Is I_n intersection with I_m is an ideal of the form of I_k?
C) Is I_n union with I_m is an ideal of the form of I_k?

I just used the Bezout's identity that d = ax+by for any d is a common devisor of a,b, and x,y are integers.
And I get that A is a proof.
For B I don't know how to start...

 

[fresh_42]

Hello Experts,

I post this question here because in the homework topics there is no abstract algebra!
Please help me I want to understand it:

I have a ring R with unit. Also I am given n - natural number, I_n is the set {x in R: n*x = 0}

I have to prove or refute: Given n, m natural numbers:

A) Is I_n + I_m is an ideal of the form of I_k?
B) Is I_n intersection with I_m is an ideal of the form of I_k?

Let's write I_n=nR$.$nR \cap mR = (l.c.m.)R## which is again an ideal of this form. Bezout in part A) gave you the greatest common divisor, now we need the least common multiple.

C) Is I_n union with I_m is an ideal of the form of I_k?

No. E.g. $2\mathbb{Z} \cup 3\mathbb{Z} = \{\,\ldots,-6,-4,-3,-2,0,2,3,4,6,\ldots\,\}$ but $2+3=5 \notin 2\mathbb{Z} \cup 3\mathbb{Z}$.

I just used the Bezout's identity that d = ax+by for any d is a common devisor of a,b, and x,y are integers.
And I get that A is a proof.

Yes, right ideals to be exact: $(nR+mR)\cdot R \subseteq nR+mR$ and $nR+mR = dR = kR$ with Bezout's lemma, correct.

For B I don't know how to start...