Dedekind-finite ring

In mathematics, a ring is said to be a Dedekind-finite ring 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.

Definitions

A ring is Dedekind-finite if any of the following equivalent conditions hold:

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.

Examples

Any Commutative rings is Dedekind-finite
Any finite ring is Dedekind-finite.
Any Matrix rings are Dedekind-finite.
Any domain is Dedekind-finite.
Any left/right Noetherian ring is Dedekind-finite.
Given a group, the group algebra is Dedekind-finite.
A Dedekind-finite ring with an idempotent implies that the corner ring is also Dedekind-finite.
A ring with finitely many nilpotents is Dedekind-finite.
A unit-regular ring is Dedekind-finite.

A counter-example can be constructed by considering the polynomial ring, where the ring has no zero divisors and the indeterminates do not commute, 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

Properties

Dedekind-finite rings are closed under subrings, direct products, and finite direct sums. 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.
A ring is Dedekind-finite if and only if so is its opposite ring. 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. 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. 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, 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.