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Is there a term for an “inverse-closed” subring of a ring?
This is a question about terminology. Is there a standard name for a subring $A \subset B$ that has the property that an element $a \in A$ is invertible in $A$ if and only if $a$ is invertible as an element of $B$? If $a$ is invertible in $A$ then it is evidently invertible in $B$, so another way of stating the desired property is that $A$ is a subring such that if $a\in A$ is invertible in $B$ then its inverse $a^{-1}$ is contained in $A$.
For example, the center of a ring always has this property.
