Addition of ideals

Question

Given a ring $R$ and ideals $A,C$ suppose we have $A + B' =A + B = C.$ I was wondering then what can we say about relation between $B$ and $B'$.

Clearly, $B$ may not equal $B'$, but can we say something? Does it follow that $B= B' + D$ where $D$ is an ideal contained in $A$? Thanks!

Also, I was wondering, is there a way to make subtraction of ideals well defined in some sense? — Tom Mosher, Jul 15 '13 at 19:33
In general no. But suppose they are ideals in a polynomial ring and you fix a monomial order. Then you can take the generators of $A$ followed by the generators of $C$. Apply Buchberger algorithm to get a Groebner basis of $C$. The generators added to the basis after the generators of $C$ begin to be used give you a $B$. It is unique after making all these choices. — OR., Jul 15 '13 at 19:46
@RGB Thank you for this information! Just from curiosity could you do something similar if you are dealing with ring of algebraic integers? — Tom Mosher, Jul 16 '13 at 15:22
Yes, but the answer is kind of trivial. Suppose $(a)+J=(b)$. The above translates here to find the greatest common divisor of $a$ and $b$, but that is $b$. So you get $J=(b)$. Notice that $b$ divides $a$ for the problem to have a solution. — OR., Jul 16 '13 at 15:30

score 2 · Accepted Answer · answered Jul 15 '13 at 19:41

In $\mathbb Z$, $(p)+(q)=(p)+(r)$ for all distinct primes $p,q,r$.

More generally, if $M$ is any maximal ideal in a ring $R$ and $I,J$ are any two ideals that are not contained in $M$, then $M+I=M+J$. So there is enormous freedom in constructing the situation you describe with no particular conditions on the ideals.

Addition of ideals

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