In mathematics, specifically abstract 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 a ∗ b = b ⋅ a for all a, b in R. 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).
Free algebra with two generatorsEdit
Then the opposite algebra has multiplication given by
which are not equal elements.
- , , and
All elements of are of the form
If the multiplication of is denoted , it has the multiplication table
Then the opposite algebra with multiplication denoted has the table
A commutative algebra is isomorphic to its opposite algebra since for all and in .
- 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.
- The left ideals of a ring are the right ideals of its opposite.
- The opposite ring of a division ring is a division ring.
- A left module over a ring is a right module over its opposite, and vice versa.
- Berrick & Keating (2000), p. 19
- Bourbaki 1989, p. 101. sfn error: no target: CITEREFBourbaki1989 (help)
- Milne. Class Field Theory. p. 120.
- Bourbaki 1989, p. 103. sfn error: no target: CITEREFBourbaki1989 (help)
- Bourbaki 1989, p. 114. sfn error: no target: CITEREFBourbaki1989 (help)
- Bourbaki 1989, p. 192. sfn error: no target: CITEREFBourbaki1989 (help)
- Berrick, A. J.; Keating, M. E. (2000). An Introduction to Rings and Modules With K-theory in View. Cambridge studies in advanced mathematics. Vol. 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.