Some results on very strong finite-type modules

In the context of commutative rings, the very strong finite-type (VSFT) condition was introduced first by J. Coykendall and T. Dutta. In this work, we introduce the notion of very strong finite type (VSFT) modules over commutative rings, extending the concept of VSFT rings to modules. We establish basic properties of VSFT modules and show that they are necessarily finitely generated. A Cohen type characterization is proved in terms of prime ideals. We study the stability of the VSFT property under direct products, homomorphic images and quotients, and give several characterizations supported by examples.