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On Unitification of $*$-rings

Sanjay More, Anil Khairnar, B. N. Waphare

Abstract

S. K. Berberian raised the open problem ``Can every weakly Rickart $*$-ring be embedded in a Rickart $*$-ring? with preservation of right projections?" Berberian has given a partial solution to this problem. Khairnar and Waphare raised a similar problem for p.q.-Baer $*$-rings and gave a partial solution. In this paper, we give more general partial solutions to both the problems.

On Unitification of $*$-rings

Abstract

S. K. Berberian raised the open problem ``Can every weakly Rickart -ring be embedded in a Rickart -ring? with preservation of right projections?" Berberian has given a partial solution to this problem. Khairnar and Waphare raised a similar problem for p.q.-Baer -rings and gave a partial solution. In this paper, we give more general partial solutions to both the problems.
Paper Structure (3 sections, 19 theorems)

This paper contains 3 sections, 19 theorems.

Table of Contents

  1. Introduction
  2. Unitification of Weakly Rickart $*$-rings
  3. Unitification of Weakly p.q.-Baer $*$-rings

Key Result

Proposition 1.1

If $R$ is Rickart $*$-ring, then $R$ has a unity element and the involution of $R$ is proper.

Theorems & Definitions (29)

  • Proposition 1.1: Ber
  • Proposition 1.2: Ber
  • Theorem 1.3: Anil
  • Proposition 1.4: Anil
  • Theorem 1.5: Anil
  • Lemma 2.1: Ber
  • Lemma 2.2: Ber
  • Theorem 2.3: Ber
  • Theorem 2.4: Tha3
  • Theorem 2.5: Tha3
  • ...and 19 more