regular elements of finite ring


If the finite ring R has regular elementsPlanetmathPlanetmathPlanetmath, then it has a unity.  All regular elements of R form a group under the ring multiplication and with identity elementMathworldPlanetmath the unity of R.  Thus the regular elements are exactly the units of the ring; the rest of the elements are the zero and the zero divisors.

Proof.  Obviously, the set of the regular elements is non-empty and closed under the multiplication.  Let’s think the multiplication table of this set.  It is a finite distinct elements (any equationax=ay  reduces to  x=y).  Hence, for every regular element a, the square a2 determines another a such that  a2a=a.  This implies  a(a2a)(aa)=aa(aa),  i.e.  (aa)(aa)2=(aa)(aa),  and since aa is regular (,  we obtain that  (aa)2=aa.  So aa is idempotentPlanetmathPlanetmath, and because it also is , it must be the unity of the ring (  aa=1.  Thus we see that R has a unity which is a regular element and that a has a multiplicative inverse a, also regular.  Consequently the regular elements form a group.

Title regular elements of finite ring
Canonical name RegularElementsOfFiniteRing
Date of creation 2013-03-22 15:11:09
Last modified on 2013-03-22 15:11:09
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 19
Author pahio (2872)
Entry type Theorem
Classification msc 13G05
Classification msc 16U60
Related topic GroupOfUnits
Related topic PrimeResidueClass
Related topic WedderburnsTheorem
Related topic Unity