$S$-کنش‌های خالص نسبت به ایده‌آل‌های چپ چگال

نویسندگان
مهدیه حدادی
سید مجتبی ناصر شیخ الاسلامی
دانشگاه سمنان
10.52547/mmr.7.3.495
چکیده
در این مقاله قصد داریم مانند آنچه را که لمبک در مورد تعمیم مفهوم زیرمدول‌های خالص با استفاده از یک رادیکال‌ در رسته $R$-مدول‌ها انجام داده است، برای رسته‌ی $S$-کنش‌ها انجام دهیم. به این منظور، ما نوع خاصی از زیرکنش‌های خالص، یعنی زیرکنش‌های $dli$-خالص، که مرتبط با مفهوم رادیکال هونکه است، را در این رسته معرفی می‌کنیم. همچنین نشان می‌دهیم که برای هر رادیکال هونکه

$S$ ،$r$-کنش‌های به طور مطلق $dli$-خالص همان $S$-کنش‌های $r$-انژکتیو ضعیف هستند. سپس به وسیله‌ی هر ایده‌آل چپ $r$-چگال، عملگر بستاری را معرفی می‌کنیم که ارتباط نزدیکی با مفهوم $S$-کنش‌های $dli$-خالص دارد. سپس این ارتباط را به‌ طور مفصل مورد بررسی قرار می‌دهیم.
کلیدواژه‌ها
S-کنش
رادیکال
ایده‌آل r-چگال
زیرکنش dli-خالص

عنوان مقاله English

On dense left ideal pure $S$-acts

نویسندگان English

mahdieh Haddadi
seyed mojtaba naser Sheykholislami
Semnan University
چکیده English

In this paper, similar to the Lembek's idea concerning a generalization of the notion of purity associated with a radical in the category of R-modules, we give the notion of purity related to a Hoehnke radical, d.l.i.pure, in the category of $S$-acts and investigate this notion. We also show that absolutely d.l.i.pure $S$-acts are exactly the $r$-weakly injective $S$-acts, for every Hoehnke radical $r$, and we give several characterization of d.l.ipure $S$-acts. We then present a closure operator induced by an $r$-dense left ideal which is closely related to the notion of d.l.i.purity and give a characterization of d.l.i.pure $S$-acts by this closure operator.

./files/site1/files/%D8%AD%D8%AF%D8%A7%D8%AF%DB%8C-%DA%86%DA%A9%DB%8C%D8%AF%D9%87__%D8%A7%D9%86%DA%AF%D9%84%DB%8C%D8%B3%DB%8C(1).pdf

کلیدواژه‌ها English

$S$-act
radical
$r$-dense
d.l.i.pure
[1] Banaschewski, B. Injectivity and essential extensions in equational classes of algebras. In Proc. Conf. on Universal Algebra (Queen’s Univ., Kingston, Ont., 1969), pages 131–147. Queen’s Univ., Kingston, Ont., 1970.
[2] Banaschewski, B. Equational compactness of G-sets. Canad. Math. Bull., 17:11–18, 1974.
[3] Banaschewski, B. and Nelson, E. Equational compactness in equational classes of algebras. Algebra Universalis, 2:152–165, 1972.
[4] Barzegar, H. Sequentially pure injectivity. Quaest. Math., 38(2):191–201, 2015.
[5] Barzegar, H. and Ebrahim, M. M. Sequentially pure monomorphisms of acts over semigroups. Eur. J. Pure Appl. Math., 1(4):41–55, 2008.
[6] Barzegar, H., Ebrahimi, M. M., and Mahmoudi, M. Essentiality and injectivity relative to sequential purity of acts. Semigroup Forum, 79(1):128–144, 2009.
[7] Cohn, P. M. On the free product of associative rings. Math. Z., 71:380–398, 1959.
[8] Ebrahimi, M. M. On ideal closure operators of M-sets. Southeast Asian Bull. Math., 30(3):439–444, 2006.
[9] Ebrahimi, M. M., Haddadi, M., and Mahmoudi, M. Injectivity in a category: an overview on well behavior theorems. Algebras Groups Geom., 26(4):451–471, 2009.
[10] Ebrahimi, M M., Haddadi, M., and Mahmoudi, M. Injectivity in a category: an overview on smallness conditions. Categ. General Alg. Struct. Appl., 2(1):83–112, 2014.
[11] Ebrahimi, M. M., Mahmoudi, M., and Shahbaz, L. Proper behaviour of sequential injectivity of acts over semigroups. Comm. Algebra, 37(7):2511–2521, 2009.
[12] Gacsályi, S. On pure subgroups and direct summands of abelian groups. Publ. Math. Debrecen, 4:89–92, 1955.
[13] Gould, V. Completely right pure monoids. Proc. Roy. Irish Acad. Sect. A, 87(1):73–82, 1987.
[14] Gould, V. Coperfect monoids. Glasgow Math. J., 29(1):73–88, 1987.
[15] Haddadi, M and Ebrahimi, MM. A radical extension of the category of s-sets. Bulletin of the Iranian Mathematical Society, 43(5):1153–1163, 2017.
[16] Haddadi, M. and Shaykholislami, S. M. N. On radical and torsion theory in the category of S-acts. Submitted for publication, 2016.
[17] Haddadi, M. and Shaykholislami, S. M. N. Radical-injectivy in the category S-act. Submitted for publication, 2017.
[18] Kilp, M., Knauer, U., and Mikhalev, A. V. Monoids, acts and categories, volume 29 of De Gruyter Expositions in Mathematics. Walter de Gruyter & Co., Berlin, 2000.
[19] Lambek, J. Torsion theories, additive semantics, and rings of quotients. Lecture Notes in Mathematics, Vol. 177. Springer-Verlag, Berlin-New York, 1971.
[20] Mahmoudi, M. and Ebrahimi, M. M. Purity and equational compactness of projection algebras. Appl. Categ. Structures, 9(4):381–394, 2001.
[21] Mahmoudi, M. and Moghaddasi, Gh. Sequential purity and injectivity of acts over some classes of semigroups. Taiwanese J. Math., 15(2):737–744, 2011.
[22] Normak, P. Purity in the category of M-sets. Semigroup Forum, 20(2):157–170, 1980.
[23] Wiegandt, R. Radical and torsion theory for acts. Semigroup Forum, 72(2):312–328, 2006.