Subring

definitionring-theory

Definition

Given a ring $(R,+,\cdot )$, $\varnothing \ne S\subseteq R$ is a subring if $(S,+,\cdot )$ is a ring with the induced operations.

Equivalently, $(S,+,\cdot )$ is a subring of $(R,+,\cdot )$ if and only if $(S,+)\le (R,+)$ and $S$ is stable under products.

Denote $S\le (R,+,\cdot )$ if $S$ is a subring.

If $R$ is unital, and $S\le R$, $1_R\in S$, we say $S$ is a unital subring (note that this is note necessary of all subrings of unital rings).

Examples