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.
Hezarjaribi, M. (2025). On $T$-injectivity and $T_ \cap$-injectivity in the category of $S$-acts. Journal of Algebra and Related Topics, (), -. doi: 10.22124/jart.2025.30715.1803
MLA
Hezarjaribi, M. . "On $T$-injectivity and $T_ \cap$-injectivity in the category of $S$-acts", Journal of Algebra and Related Topics, , , 2025, -. doi: 10.22124/jart.2025.30715.1803
HARVARD
Hezarjaribi, M. (2025). 'On $T$-injectivity and $T_ \cap$-injectivity in the category of $S$-acts', Journal of Algebra and Related Topics, (), pp. -. doi: 10.22124/jart.2025.30715.1803
CHICAGO
M. Hezarjaribi, "On $T$-injectivity and $T_ \cap$-injectivity in the category of $S$-acts," Journal of Algebra and Related Topics, (2025): -, doi: 10.22124/jart.2025.30715.1803
VANCOUVER
Hezarjaribi, M. On $T$-injectivity and $T_ \cap$-injectivity in the category of $S$-acts. Journal of Algebra and Related Topics, 2025; (): -. doi: 10.22124/jart.2025.30715.1803
)