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Can anyone please give some examples or give a reference where I can find examples of non-commutative rings with no multiplicative identity other than matrix rings ? Also examples of finite non-commutative rings and finite rings with no multiplicative identity ?

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Group algebras for nonabelian groups are noncommutative.

A semi group without identity would create a semigroup algebra without identity.

If the field and the semi group are finite, so I'd the semigroup ring.

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