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Dedekind-finite ring
Mathematical concept From Wikipedia, the free encyclopedia
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In mathematics, a ring is said to be a Dedekind-finite ring (also called directly finite rings[1][2][3] and Von Neumann finite rings[4][2][3]) if ab = 1 implies ba = 1 for any two ring elements a and b. In other words, all one-sided inverses in the ring are two-sided. Numerous examples of Dedekind-finite rings include Commutative rings, finite rings, and Noetherian rings.
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Definitions
A ring is Dedekind-finite if any of the following equivalent conditions hold:[3][better source needed]
- All one sided inverses are two sided: implies .
- Each element that has a right inverse has a left inverse: For , if there is a where , then there is a such that .
- Capacity condition: , implies .
- Each element has at most one right inverse.
- Each element that has a left inverse has a right inverse.
- Dual of the capacity condition: , implies .
- Each element has at most one left inverse.
- Each element that has a right inverse also has a two sided inverse.
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Examples
- Any Commutative rings is Dedekind-finite[3]
- Any finite ring is Dedekind-finite.[4]
- Any Matrix rings are Dedekind-finite.[3]
- Any domain is Dedekind-finite.[4]
- Any left/right Noetherian ring is Dedekind-finite.[4][3]
- Given a group , the group algebra is Dedekind-finite.[3]
- A Dedekind-finite ring with an idempotent implies that the corner ring is also Dedekind-finite.[2]
- A ring with finitely many nilpotents is Dedekind-finite.[3]
- Reversible rings, rings where must imply , are Dedekind-finite.[3]
- A unit-regular ring is Dedekind-finite.[4]
- A ring without left/right zero divisors is Dedekind-finite.[2]
A counter-example can be constructed by considering the polynomial ring , where the ring has no zero divisors and the indeterminates do not commute (that is, ), being divided by the ideal , then has a right inverse but is not invertible. This illustrates that Dedekind-finite rings need not be closed under homomorpic images[2]
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Properties
Dedekind-finite rings are closed under subrings[1][2][better source needed], direct products[3][2], and finite direct sums.[2] This makes the class of Dedekind-finite rings a Quasivariety, which can also be seen from the fact that its axioms are equations and the Horn sentence .[2]
A ring is Dedekind-finite if and only if so is its opposite ring.[2] If either a ring , its polynomial ring with indeterminates , the free word algebra over with coefficients in , or the power series ring are Dedekind-finite, then they all are Dedekind-finite.[2] Letting denote the Jacobson radical of the ring , the quotient ring is Dedekind-finite if and only if so is , and this implies that local rings and semilocal rings are also Dedekind-finite.[2] This extends to the fact that, given a ring and a nilpotent ideal , the ring is Dedekind-finite if and only if so is the quotient ring ,[2] and as a consequence, a ring is also Dedekind-finite if and only if the upper triangular matrices with coeffecients in the ring also form a Dedekind-finite ring.[2]
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References
See also
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