Subring Tests

The following theorems justify the ability to check that a subset of another ring is a subring without checking all axioms.

Theorem

Given a non-unital ring \((R, +, \cdot)\) and subset of \(S \subseteq R\), \(S\) is a non-unital subring of \(R\) if and only if for all \(a, b \in S\),

  1. \(S \neq \varnothing\)
  2. \(a - b \in S\)
  3. \(ab \in S\)
Proof

The fact that \(S\) is an additive subgroup of \(R\) follows from the corresponding subgroup test and properties 1 and 2.

We then have closure under multiplication by assumption, and associativity of multiplication and distributivity of multiplication across addition follow from restricting these operations from the main group to this subset, with well definedness of these restrictions already proven.

The converse is trivial.

Corollary

Given a ring with identity \((R, +, \cdot)\) and subset of \(S \subseteq R\), \(S\) is a subring with identity of \(R\) if and only if for all \(a, b \in S\),

  1. \(1 \in S\)
  2. \(a - b \in S\)
  3. \(ab \in S\)
Proof

Clearly if \(1 \in S\), then \(S \neq \varnothing\). By the previous result \(S\) is a non-unital subring of \(R\). Then because \(1 \in S\), \(S\) must be a subring with identity.

Simpler versions of these results exist for finite subsets just as in the case of groups.