Opposite ring
Encyclopedia
In algebra
, the opposite of a ring
is another ring with the same elements and addition operation, but with the multiplication performed in the reverse order.
More precisely, the opposite of a ring (R, +, ·) is the ring (R, +, *), whose multiplication '*' is defined by a * b = b · a. (Ring addition is per definition always commutative.)
if, and only if, its opposite is commutative. If two rings R1 and R2 are isomorphic, then their corresponding opposite rings are also isomorphic. The opposite of the opposite of a ring is isomorphic to that ring. A ring and its opposite ring are anti-isomorphic.
A commutative ring is always equal to its opposite ring. A non-commutative ring may or may not be isomorphic to its opposite ring.
Algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
, the opposite of a ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
is another ring with the same elements and addition operation, but with the multiplication performed in the reverse order.
More precisely, the opposite of a ring (R, +, ·) is the ring (R, +, *), whose multiplication '*' is defined by a * b = b · a. (Ring addition is per definition always commutative.)
Properties
A ring (R, +, · ) is commutativeCommutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
if, and only if, its opposite is commutative. If two rings R1 and R2 are isomorphic, then their corresponding opposite rings are also isomorphic. The opposite of the opposite of a ring is isomorphic to that ring. A ring and its opposite ring are anti-isomorphic.
A commutative ring is always equal to its opposite ring. A non-commutative ring may or may not be isomorphic to its opposite ring.