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Let $\struct {R, +, \circ}$ be an algebraic structure with two operations.

A subring of $\struct {R, +, \circ}$ is a subset $S$ of $R$ such that $\struct {S, +_S, \circ_S}$ is a ring.

Proper Subring

A subring $S$ of $R$ is a proper subring of $R$ if and only if $S$ is neither the null ring nor $R$ itself.

Also defined as

Sources which deal only with rings with unity typically demand that the unity is also part of a subring.

Some sources insist that $R$ must be a ring for $S$ to be definable as a subring, but this limitation is unnecessarily restricting.

Also see

  • Results about subrings can be found here.