In the context of commutative rings, the VSFT (very strong finite type) 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.
Dabbabi et al. (Fri,) studied this question.