No conmmutative rings examples I want to met more examples for commutative and noncommutative rings without an element unique for multiplication.
As the set of all even integers (commutative) $2\mathbb{Z}$ or the set of matrices with entries in the even integers (non-commutative) $M_{n}(2\mathbb{Z})$.

Take your favorite non commutative ring $R$ with identity and consider $S$, the direct sum of infinitely many copies of $R$, with pointwise addition and multiplication. This ring is non commutative and has no identity element.
Take your favorite non commutative ring $R$ with identity and consider a proper right ideal $I$ thereof.
If $I$, considered as a ring has an identity $e$, then this is an idempotent element and so is $1-e$; moreover $I=eR$. Consider the right ideal $J=(1-e)R$. Then $R\cong I\oplus J$ as right $R$-modules, via the map $r\mapsto(er,(1-e)r)$.
So if your ring $R$ is indecomposable as right $R$-module, then $I$ is a ring without identity.
