No abstract

No abstract

In a ring with involution we already know that the Moore Penrose inverse can be used to construct the properties of normal elements. By using the fact that the group inverse of an element in a ring will be commutative with element which is commutative with that element, this paper explains that the generalization of Moore Penrose inverse can be also to establish some properties of normal elements in a ring with an involution. Some elements in the ring such as symmetric, EP, partial isometries, etc are also can be expressed in group inverse. So the results of this paper much be required for built the properties of those elements.

The definition of symmetric elements in a ring equipped with involution can be generalized by multiplying it as many as n natural number. Generalizing of symmetric elements is known as the generalized symmetric elements. Not each generalized symmetric element has a generalized Moore Penrose inverse. That condition is an extension of the symmetric elements properties in a ring. This paper produces others properties of generalized symmetric elements. Method which is used is by generalizing the commutative properties between symmetric elements with generalized Moore Penrose inverse.

In this paper, we present several new characteristics of the Moore-Penrose inverse in rings with involution. We use the concept of the Drazin inverse to build these characteristics in purely algebraic terms. We only discuss on the symmetric element of Moore-Penrose invertible.