Opposite ring

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 explicitly, the opposite of a ring (R, +, ·) is the ring (R, +, ∗) whose multiplication ∗ is defined by ab = b · a.[1][2] The opposite ring can be used to define multimodules, a generalization of bimodules. They also help clarify the relationship between left and right modules (see #Properties).

Monoids, groups, rings, and algebras can be viewed as categories with a single object. The construction of the opposite category generalizes the opposite group, opposite ring, etc.

Properties

  • Two rings R1 and R2 are isomorphic if and only if their corresponding opposite rings are isomorphic
  • The opposite of the opposite of a ring R is isomorphic to R.
  • A ring and its opposite ring are anti-isomorphic.
  • A ring is commutative if and only if its operation coincides with its opposite operation.[2]
  • The left ideals of a ring are the right ideals of its opposite.[3]
  • The opposite ring of a field is a field (regardless of whether the field is commutative).[4]
  • A left module over a ring is a right module over its opposite, and vice versa.[5]

Notes

  1. Berrick & Keating (2000), p. 19
  2. Bourbaki 1989, p. 101.
  3. Bourbaki 1989, p. 103.
  4. Bourbaki 1989, p. 114.
  5. Bourbaki 1989, p. 192.

References

  • Berrick, A. J.; Keating, M. E. (2000). An Introduction to Rings and Modules With K-theory in View. Cambridge studies in advanced mathematics. 65. Cambridge University Press. ISBN 978-0-521-63274-4.
  • Nicolas, Bourbaki (1989). Algebra I. Berlin: Springer-Verlag. ISBN 978-3-540-64243-5. OCLC 18588156.

See also

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