Opposite ring

The counter ring to a ring is a construction of the mathematical subfield of ring theory. The counter-ring into a ring is formed, characterized in that one exchanges the factors in the multiplication.

Definition

It it a ring. Then the counter-ring is ( engl. opposite ring ) defined as follows:

• The underlying set of is.
• The addition to match with that on.
• The multiplication is defined by the multiplication of the following: all.

Is therefore substantially equal to the output ring, only when the multiplication factor of the order will be reversed relative to the output ring.

Properties

• Is commutative, then obviously.
• Sentences about left ideals in a ring are sets of right ideals. Therefore rates that apply for all left ideals in all rings, also apply to right ideals in all rings.
• Is algebra over a field, it is also such an algebra, by using for the same vector space structure. We also speak of the opposite algebra.
• It is the algebra of matrices over a field. Then for the transposition known the rule. This means that the transposition is a ring homomorphism, even an isomorphism. More generally, a Antihomomorphismus between two rings is a homomorphism and
• In general and are not isomorphic. Examples can be found where certain left-right symmetries do not apply. There are, for example linksnoethersche rings that are not rechtsnoethersch; such rings can not be isomorphic to its opposite rings.
• Is-a links module, then by the definition of a -module.