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Here is a criterion for group

Let $G$ be a group and $H,K$ be subgroups of $G$.

Then, $HK$ is a subgroup of $G$ iff $HK=KH$.

Just like this, I'm curious to know whether there is a similar criterion for rng

That is:

Let $R$ be an rng and $S,T$ be subrngs of $R$.

Is there a criterion for $S+T$ to be a subrng?

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cococomi
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  • The fact that there are two operations makes it much less likely. For example, if $$R=\mathbb{C}[x,y],\quad S=\mathbb{C}[x],\quad T=\mathbb{C}[y]$$ then $S+T$ is not a subring because it's not closed under multiplication. – curious Nov 08 '14 at 17:58

1 Answers1

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In order for $S+T$ to be a subring we must have that for every $s,s'\in S$ and $t,t'\in T$,

$$(s+t)(s'+t')=ss'+st'+ts'+tt'\in S+T$$

Since $ss'+tt'\in S+T$ and $S+T$ is closed under addition, the requirement is equivalent to saying that $st'+ts'\in S+T$ for all $s,s'\in S$ and $t,t'\in T$. Thus $S+T$ is a subring if and only if $$ST+TS\subseteq S+T$$

Matt Samuel
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