Definition of a Subring

Let $(R,+,\cdot )$ be a ring. We say that a subset $S\subset R$ is a subring if $(S,+,\cdot )$ is itself a ring.

Notation: $S\le R$.

Quick way to test a subset:

Definition of Quick Subring Theorem

A subset $S$ of a ring $R$ is a subring if and only if…

1. $S$ is non-empty.
2. $S$ is closed under both addition and multiplication of $R$.
3. $S$ contains the negative (i.e. the additive inverse) of each of its elements.

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