Öz
This paper investigates relative ring theoretical properties in the context of formal triangular matrix rings. The first aim is to study the semiregularity of formal triangular matrix rings relative to an ideal. We prove that the formal triangular matrix ring $T$ is $T'$-semiregular if and only if $A$ is $I$-semiregular, $B$ is $K$-semiregular and $N=M$ for an ideal $T'=\bigl(\begin{smallmatrix}
I & 0\\
N & K
\end{smallmatrix}\bigr)$ of $T=\bigl(\begin{smallmatrix}
A & 0\\
M & B
\end{smallmatrix}\bigr).$ We also discuss the relative semiperfect formal triangular matrix rings in relation to the strong lifting property of ideals. Moreover, we have considered the behavior of relative semipotent and potent property of formal triangular matrix rings. Several examples are provided throughout the paper in order to highlight our results.
Anahtar Kelimeler
Formal triangular matrix ring strongly lifting ideal semiregular ring semiperfect ring semipotent ring
Kaynakça
- Altun Özarslan, M, Lifting properties of formal triangular matrix rings, RACSAM, 115(3) (2021), Paper No. 104, 13 pp. https://doi.org/10.1007/s13398-021-01044-0
- Berberian, S. K., The center of a corner of a ring, J. Algebra, 71(2) (1981), 515-523. https://doi.org/10.1016/0021-8693(81)90191-5
- Goodearl, K. R., Ring Theory. Nonsingular Rings and Modules, Monographs and Textbooks in Pure and Appl. Math., 33, Marcel Dekker, New York, 1976.
- Haghany, A, Varadarajan, K., Study of formal triangular matrix rings, Comm. Algebra, 27 (1999), 5507-5525. https://doi.org/10.1080/00927879908826770
- Herstein, I. N., A counter example in Noetherian rings, Proc. Nat. Acad. Sci. U.S.A., 54 (1965), 1036-1037. https://doi.org/10.1073/pnas.54.4.1036
- Hong, C. Y., Kim, N. K., Lee, Y., Exchange rings and their extensions, J. Pure Appl. Algebra, 179 (2003), 117-126. https://doi.org/10.1016/S0022-4049(02)00299-2
- Nicholson, W. K., Lifting idempotents and exchange rings, Trans. Amer. Math. Soc., 229 (1977), 269-278. https://doi.org/10.1090/S0002-9947-1977-0439876-2
- Nicholson, W. K., Yousif, M.F., Weakly continuous and C2-rings, Comm. Algebra, 29 (2001), 2429-2446. https://doi.org/10.1081/AGB-100002399
- Nicholson, W. K., Zhou, Y, Strong lifting, J. Algebra, 285(2) (2005), 795-818. https://doi.org/10.1016/j.jalgebra.2004.11.019
- Yousif, M. F., Zhou, Y, Semiregular, semiperfect and perfect rings relative to an ideal, Rocky Mountain J. Math., 32 (2002), 1651-1671. https://doi.org/10.1216/rmjm/1181070046
Öz
Kaynakça
- Altun Özarslan, M, Lifting properties of formal triangular matrix rings, RACSAM, 115(3) (2021), Paper No. 104, 13 pp. https://doi.org/10.1007/s13398-021-01044-0
- Berberian, S. K., The center of a corner of a ring, J. Algebra, 71(2) (1981), 515-523. https://doi.org/10.1016/0021-8693(81)90191-5
- Goodearl, K. R., Ring Theory. Nonsingular Rings and Modules, Monographs and Textbooks in Pure and Appl. Math., 33, Marcel Dekker, New York, 1976.
- Haghany, A, Varadarajan, K., Study of formal triangular matrix rings, Comm. Algebra, 27 (1999), 5507-5525. https://doi.org/10.1080/00927879908826770
- Herstein, I. N., A counter example in Noetherian rings, Proc. Nat. Acad. Sci. U.S.A., 54 (1965), 1036-1037. https://doi.org/10.1073/pnas.54.4.1036
- Hong, C. Y., Kim, N. K., Lee, Y., Exchange rings and their extensions, J. Pure Appl. Algebra, 179 (2003), 117-126. https://doi.org/10.1016/S0022-4049(02)00299-2
- Nicholson, W. K., Lifting idempotents and exchange rings, Trans. Amer. Math. Soc., 229 (1977), 269-278. https://doi.org/10.1090/S0002-9947-1977-0439876-2
- Nicholson, W. K., Yousif, M.F., Weakly continuous and C2-rings, Comm. Algebra, 29 (2001), 2429-2446. https://doi.org/10.1081/AGB-100002399
- Nicholson, W. K., Zhou, Y, Strong lifting, J. Algebra, 285(2) (2005), 795-818. https://doi.org/10.1016/j.jalgebra.2004.11.019
- Yousif, M. F., Zhou, Y, Semiregular, semiperfect and perfect rings relative to an ideal, Rocky Mountain J. Math., 32 (2002), 1651-1671. https://doi.org/10.1216/rmjm/1181070046
Ayrıntılar
| Birincil Dil | İngilizce |
|---|---|
| Konular | Matematik |
| Bölüm | Research Article |
| Yazarlar | |
| Yayımlanma Tarihi | 16 Mart 2024 |
| Gönderilme Tarihi | 30 Mayıs 2023 |
| Kabul Tarihi | 5 Kasım 2023 |
| Yayımlandığı Sayı | Yıl 2024 Cilt: 73 Sayı: 1 |
Kaynak Göster
Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics.
