Let $R$ be a commutative ring.
Then, is it possible to extend this to have a unity?
That is, is there a commutative ring with unity $R'$ such that $R$ is a subring of $R'$?
2026-03-25 03:00:50.1774407650
Is it possible to extend a commutative ring to have a unity?
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Construct a ring $S$ having pairs $\left(n,r\right)$ as elements where $n\in\mathbb{Z}$ and $r\in R$ as follows:
Define addition by $\left(n,r\right)+\left(m,s\right)=\left(n+m,r+s\right)$
Define multiplication by $\left(n,r\right)\left(m,s\right)=\left(nm,mr+ns+rs\right)$.
It can be shown easily that $S$ equipped with this addition and multiplication is a ring with identity $(1,0_R)$. The original ring $R$ can be embedded in $S$ by $r\mapsto (1,r)$
If $R$ is commutative then so is $S$.