In mathematics, a ring is said to be a Dedekind-finite ring (also called directly finite rings and Von Neumann finite rings) 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.
A ring is Dedekind-finite if any of the following equivalent conditions hold:
A counter-example can be constructed by considering the free algebra (a "polynomial ring" in two non-commuting indeterminates, that is, ), where the ring has no zero divisors, 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.
Another non-example is the endomorphism ring of a vector space (or free module) with a countably infinite basis . Let be the left shift defined by and for , and let be the right shift . Then , but .
A matrix ring over a Dedekind-finite ring may also fail to be Dedekind-finite. For this, one can consider , where is a field, let , , and the two-sided ideal of generated by the entries of . Then is a domain, but in .
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.