Ring with Identity
Definition
A ring with identity \(R\) is a non-unital ring in which there is a multiplicative identity.
That is, there is an element, often denoted by \(1\), such that:
\[ 1 \times a = a \times 1 = a\]
for all \(a \in R\)
This means a non-unital ring is an abelian group under addition and a monoid under multiplication, with the additional requirement that multiplication is left and right distributive over addition.