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On $T$-injectivity and $T_ \cap$-injectivity in the category of $S$-acts | ||
| Journal of Algebra and Related Topics | ||
| مقالات آماده انتشار، اصلاح شده برای چاپ، انتشار آنلاین از تاریخ 22 آذر 1404 اصل مقاله (185.1 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22124/jart.2025.30715.1803 | ||
| نویسنده | ||
| M. Hezarjaribi* | ||
| Department of Mathematics, Payame Noor University, Tehran, Iran | ||
| چکیده | ||
| In this paper, we introduce and investigate new notions of injectivity and essentiality for right $S$-acts, defined relative to a multiplicatively closed subset $T$ of a monoid $S$. We study the concepts of $T$-injective and $T_\cap$-injective $S$-acts, along with $T$-essential and $T_\cap$-essential subacts. We first establish foundational definitions and illustrate the differences between $T$-essential and $T_\cap$-essential subacts with examples. Our study shows that $T$-injectivity does not necessarily imply the existence of a zero element in the $S$-act, which contrasts with classical results on injective $S$-acts. We proved that every $S$-act admits a $T_\cap$-injective hull, satisfying a universal property analogous to classical injective envelopes. We study closure properties of the classes of $T$-injective and $T_\cap$-injective $S$-acts under categorical constructions such as products, retracts, and direct limits. Moreover, we demonstrate that pushouts preserve $T_\cap$-essential extensions, while pullbacks may not, highlighting an asymmetry in categorical behavior. | ||
| کلیدواژهها | ||
| Multiplicatively closed subset؛ S-act؛ $T$-injectivity؛ $T_\cap$-injectivity | ||
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آمار تعداد مشاهده مقاله: 2 تعداد دریافت فایل اصل مقاله: 2 |
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