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given two elements $r$,$s$ in a ring $R$, are the following two notations equivalent?

  • $(r,s)$
  • $(r)+(s)$

For example, in the ring $\mathbb{Z}[X]$, is $(2,X)=(2)+(X)$?

Thanks a lot.

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up vote 11 down vote accepted

Assuming you mean $R$ to be commutative or ideal to be two-sided: Yes, it is: by the definition, $(r,s)=\{rx+sy|x,y\in R\}$.
On the other hand, $(r)+(s)=\{ax+by|a\in(r),b\in(s),x,y\in R\}$, which is the same since $a\in(r)$ implies that $a=r\alpha$.

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Looks like the definition of $(r,s)$ has an obvious typo, I'd correct it but it's too few characters so can you fix it? – Squirtle Jun 2 '14 at 0:06

By definition, $(r,s)$ is the smallest ideal containing both $r$ and $s$.

By definition, $(r)+(s)$ is the sum of the ideals generated by $r$ and $s$.

That these two are equal is actually an easy theorem.

More generally, if $A$ and $B$ are subsets of $R$ then $(A,B)$ or $\langle A,B\rangle$ denotes the smallest ideal of $R$ containing both $A$ and $B$. If $I$ and $J$ are two ideals of $R$, then $I+J$ denotes the set $\{ i + j : i\in I, j\in J\}$. It is an easy theorem that $I+J=\langle I,J\rangle$ and so $\langle A,B\rangle = \langle A \rangle + \langle B\rangle$.

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