Let Ri be commutative rings, Si multiplicatively closed subsets of Ri, and Mi unitary Ri-modules (i = 1, 2). We study the localization of the product ring R1 × R2 and the product module M1 × M2 at S1 × S2. We describe the localization map, its kernel and saturation, and establish the natural isomorphisms with the products of the factor localizations. We then examine extension and contraction of ideals and submodules, exactness, quotient and colon formulas, and preservation of multiplication modules. Finally, using contraction and saturation, we define the induced notion of ϕ-primality and prove transfer and converse results for ideals and submodules. Examples clarify the role of the hypotheses, including finite generation in the colon formula and the properness condition in the transfer theorem.
Moslemi Koopaei, M. Hosein. (2027). Localization of product rings and modules: extension, contraction, and ϕ-primality. Journal of Frame and Matrix Theory, 3(1), 37–72. https://doi.org/10.22034/jfmt.2026.564658.1032
MLA
Moslemi Koopaei, M. Hosein. "Localization of product rings and modules: extension, contraction, and ϕ-primality", Journal of Frame and Matrix Theory, 3, 1, 2027, 37–72. doi: 10.22034/jfmt.2026.564658.1032
HARVARD
Moslemi Koopaei, M. Hosein. (2027). 'Localization of product rings and modules: extension, contraction, and ϕ-primality', Journal of Frame and Matrix Theory, 3(1), pp. 37–72. doi: 10.22034/jfmt.2026.564658.1032
CHICAGO
Moslemi Koopaei, M. Hosein, "Localization of product rings and modules: extension, contraction, and ϕ-primality." Journal of Frame and Matrix Theory, 3 1 (2027): 37–72, doi: 10.22034/jfmt.2026.564658.1032
VANCOUVER
Moslemi Koopaei M. Hosein. Localization of product rings and modules: extension, contraction, and ϕ-primality. JFMT. 2027;3(1):37–72. doi: 10.22034/jfmt.2026.564658.1032
