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A ring is non-commutative if multiplication is not necessarily commutative.
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The set
of all endomorphisms of an Abelian Group forms a ring where addition is homomorphism addition and multiplication is homomorphism composition We call this ring the Endomorphism Ring.
An Endomorphism ring need not be commutative.
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A Group Ring is not necessarily commutative. In fact if
is not Abelian, then the group ring is not commutative. -
The Quaternions give a noncommutative division ring under addition and multiplication.
In fact (Fraleigh 24.9) They form a strictly skew field