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of the noncommutative rings, including the properties that apply also to commutative rings.

Sometimes the term noncommutative ring is used instead of ring to refer to an unspecified ring which is not necessarily commutative, and hence may be commutative. Generally, this is for emphasizing that the studied properties are not restricted to commutative rings, as, in many contexts, ring is used as a shorthand for commutative ring.

Although some authors do not assume that rings have a multiplicative identity, in this article we make

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