Conditions for a subset of a ring to be a subring

Dependencies:

1. Ring
2. Condition for a subset to be a subgroup

Let $S$ be a subset of a ring $R$.

$S$ is a subring iff all of the conditions below hold:

• $S$ is not the empty set.
• $\forall s_1, s_2 \in S, s_1 - s_2 \in S$.
• $\forall s_1, s_2 \in S, s_1s_2 \in S$.

Proof

Proof of 'only-if' part

If $S$ is a ring, then

• $0 \in S$, so $S$ is not empty.
• $-s_2 \in S$ since it's the additive inverse of $s_2$. $s_1 - s_2 \in S$ by closure of addition.
• $s_1s_2 \in S$ by closure of multiplication.

Proof of 'if' part

The first 2 conditions imply that $S$ is an abelian group under addition.

The third condition implies closure of multiplication.

Associativity of multiplication and distributivity are inherited from $R$.

Therefore, $S$ is a ring.

Dependency for:

1. Principal ideal

Info:

• Depth: 4
• Number of transitive dependencies: 7

Transitive dependencies:

1. Group
2. Ring
3. Identity of a group is unique
4. Subgroup
5. Inverse of a group element is unique
6. Conditions for a subset to be a subgroup
7. Condition for a subset to be a subgroup