Star, left-star, and right-star partial orders in Rickart ∗-rings

Abstract

Let be a unital ring admitting involution. We introduce an order on and show that in the case when is a Rickart -ring, this order is equivalent to the well-known star partial order. The notion of the left-star and the right-star partial orders is extended to Rickart -rings. Properties of the star, the left-star and the right-star partial orders are studied in Rickart -rings and some known results are generalized. We found matrix forms of elements and when , where is one of these orders. Conditions under which these orders are equivalent to the minus partial order are obtained. The diamond partial order is also investigated.

Keywords:

• Rickart ∗-ring
• star partial order
• left-star partial order
• right-star partial order
• diamond partial order
• minus partial order

AMS Subject Classifications:

• 16W10
• 47A05

Acknowledgments

The authors wish to thank the referee for helpful comments and suggestions, and Professor Alexander E. Guterman for proposing to study the diamond partial order.

Notes

The second and third author are supported by the Ministry of Education, Science and Technological Development of Serbia [grant no. 174007].