As the title, is a subring of a commutative ring always itself also a commutative ring?
2026-05-05 08:35:23.1777970123
Is the subring of a commutative ring always itself also a commutative ring?
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Of course. Suppose $A\subseteq B$ is a subring, and let $\cdot_A$ and $\cdot_B$ denote the multiplication operations in $A$ and $B$ respectively. Then for any $x,y\in A$, $x\cdot_A y = x\cdot_B y = y\cdot_B x = y\cdot_A x$.
I've addressed why $A$ is commutative. If that wasn't where your confusion arose, you should give your definition of subring.