A QUESTION ON *-REGULAR RINGS
2
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
A $*$-ring $R$ is called $*$-regular if every principal one-sided ideal of $R$ is generated by a projection. In this note, several characterizations of $*$-regular rings are provided. In particular, it is shown that a matrix ring $M_n(R)$ is $*$-regular if and only if $R$ is regular and $1+x_1^*x_1+\cdots+x_{n-1}^*x_{n-1}$ is a unit for all $x_i$ of $R;$ which answers a question raised in the literature recently.
Abstract
A $*$-ring $R$ is called $*$-regular if every principal one-sided ideal of $R$ is generated by a projection. In this note, several characterizations of $*$-regular rings are provided. In particular, it is shown that a matrix ring $M_n(R)$ is $*$-regular if and only if $R$ is regular and $1+x_1^*x_1+\cdots+x_{n-1}^*x_{n-1}$ is a unit for all $x_i$ of $R;$ which answers a question raised in the literature recently.
참고문헌 (8)
- S. K. Berberian, "Baer *-Rings" (1972), Springer-Verlag
- G. F. Birkenmeier, "Triangular matrix repre-sentations", J. Algebra, 230(2), pp.558~595 (2000)[DOI]
- J. Cui, "Some characterizations of *-regular rings", Comm. Algebra, 45(2), pp.841~848 (2017)[DOI]
- K. R. Goodearl, "von Neumann Regular Rings, Monographs and Studies in Mathematics, 4" (1979), Pitman (Advanced Publishing Program)
- R. Z. Han, "Generalized inverses of matrices over rings", Chinese Quart. J. Math., 7(4), pp.40~47 (1992)
- J. J. Koliha, "Moore-Penrose inverse in rings with involution", Linear Algebra Appl., 426(2-3), pp.371~381 (2007)[DOI]
- R. Penrose, "A generalized inverse for matrices", Proc. Cambridge Philos. Soc., 51, pp.406~413 (1955)[DOI]
- A. Tuganbaev, "Rings Close to Regular, Mathematics and its Applications, 545" (2002), Kluwer Academic Publishers