My professor defines a ring as a triple $(A, +,\ \cdot)$ such that $(A, +, 0)$ is an abelian group, $(A,\ \cdot)$ is a semigroup and $\cdot$ distributes over $+$. Subsequently, $B\subseteq A$ is said to be a subring of $A$ if:
- $(B, +, 0)$ is a subgroup of $(A, +, 0)$
- $(B,\ \cdot)$ is a semigroup
- if $(A,\ \cdot, 1)$ is a monoid, then $(B,\ \cdot, 1)$ is too.
If $A$ is not unitary, let's agree that the third condition holds; I guess that's the only possible interpretation. Thus, we can observe that, by the above definitions:
- a unitary ring isn't necessarily a subring of a unitary ring
- a unitary subring of a unitary ring has its same unity
- a subring of a non-unitary ring is simply a ring contained in it
- $\{0\}$ is a subring of any non-unitary ring
- the only unitary ring that contains $\{0\}$ as a subring is $\{0\}$ itself.
It appears to me that, even though there is no standard in this context, usually the third condition in the ring definition is left out when ring does not implicitly mean unitary ring. So my question is: what are the advantages of such choices? What, instead, the disadvantages?