Everything about the subring test is straightforward from the subgroup test, but the multiplicative operation of the subring, S, of ring, R, needs to be closed wrt multiplication, *. How do you prove S is closed wrt * if the only assumption about * is associativity and distributivity over addition in R? Please let me know if I need to clarify. EDIT: I found the answer. Apparently * is assumed to be closed in a ring. My algebra book made no mention of this and it seems many sources don't! The ring axioms on proofwiki.org assume * is closed in a ring, thus a subring, making my question easy to answer.