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.
These rings have also been called directly finite rings
and von Neumann finite rings.
Properties
* If a ring is finite, then it is Dedekind-finite.
* Any
subring
In mathematics, a subring of ''R'' is a subset of a ring that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and which shares the same multiplicative identity as ''R''. For those ...
of a Dedekind-finite ring is Dedekind-finite.
* If a ring is a
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
** Domain of definition of a partial function
** Natural domain of a partial function
** Domain of holomorphy of a function
* ...
then it is Dedekind-finite.
* Any
left Noetherian ring is Dedekind-finite.
* A
unit-regular ring is Dedekind-finite.
* A
local ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic n ...
is Dedekind-finite.
References
{{Reflist
See also
*
Dedekind-infinite set
In mathematics, a set ''A'' is Dedekind-infinite (named after the German mathematician Richard Dedekind) if some proper subset ''B'' of ''A'' is equinumerous to ''A''. Explicitly, this means that there exists a bijective function from ''A'' onto ...
*
Von Neumann regular ring
In mathematics, a von Neumann regular ring is a ring ''R'' (associative, with 1, not necessarily commutative) such that for every element ''a'' in ''R'' there exists an ''x'' in ''R'' with . One may think of ''x'' as a "weak inverse" of the eleme ...
Ring theory