Minimal linear codes have garnered significant attention in cryptography due to their essential role in secretsharing schemes, multiparty computation (MPC), and secure communication. While earlier studies primarily focused on minimal codes over finite fields, extending these codes to rings offers enhanced security, flexibility, and efficiency for various cryptographic applications. This article deals with minimal linear codes over finite commutative rings. Specifically, we handle a key question in minimal linear codes, which is determining the existence of an [m,k] minimal linear code withk≤m. In 2021, Lu,Wu and Cao proved that for some integerm(k;q), a minimal linear code of lengthmand dimensionkalways exists whenm≥m(k;q), providing upper and lower bounds form(k;q). This paper expands on these results by establishing both upper and lower bounds form(k;pl) andm(k;p1p2) over Zpland Zp1p2, respectively. We also present a necessary and sufficient condition for a linear code to achieve minimality when analyzed over the ring Zn. The fundamental question regarding minimal linear codes is whether a [m,k] minimal linear code exists, withkbeing less than or equal tom. Lu et al. demonstrated that there is a positive integerm(k;q) such that ifm≥m(k;q), a minimal linear code of lengthmand dimensionkover the finite field Fqmust exist (whereqis a prime power). They provided both upper and lower bounds form(k;q). In our study, we investigate the existence of minimal codes within an extended novel framework. We analyze the case of one-dimensional minimal codes over Znin depth and present improved upper bounds onm(k;q) specifically for the case whenk= 1.