Locally recoverable codes (LRCs) play a vital role in distributed storage systems where the failure or unavailability of storage devices is a common occurrence. The purpose of LRCs is to facilitate the repair processes required to recover lost or damaged data in such systems. A code C will be said (r, δ)-LRC if for each i, the ith component of codewords have locality (r, δ), that is, there exists a punctured subcode of C with support containing i, whose length is at most r + δ-1, and whose minimum distance is at least δ. An (r, δ)-LRC with locality (r, δ) allows for the local recovery of any δ-1 nodes by accessing information from r other nodes. In this paper, we present new constructions of (r, δ)-LRCs, with 2 ≤ δ ≤ p-1/t over Zps, where t divides p-1 and t ≠ p - 1. Initially, we provide generator matrices for (r, 2)-LRCs, among which one instance is considered as Singleton-Type Bound (STB)-optimal, a notion introduced in this paper. Also, we present a method for recovering an erased symbol in a codeword of our (r, 2)-LRC. For this aim, we use the polynomial interpolation over Zpsproposed by Gopalan. Next, we present the parity-check matrices for another family of (r, δ)-LRCs over Zps, and construct two instances of STB-optimal (r, δ)-LRCs. To the best of our knowledge, this paper presents the first study on ring-based LRCs. The proposed LRCs over Zpsexhibit certain design restrictions compared to LRCs over Fps. However, we provide two advantages for LRCs over Zps. First, we analyze Boolean circuits for arithmetic operations and demonstrate that the complexity of implementing multiplication in Zps, the operation with the highest cost in our algorithms, is considerably lower than in Fps. This highlights the superior performance of our LRCs in terms of implementation speed and cost-efficiency compared to their counterparts. Next, we offer an example illustrating that the Gray image of particular Zps+1-LRCs of lengthnresults in LRCs of lengthnpsover Fp, which may not necessarily be linear. This introduces a novel class of LRCs, prompting further exploration of the connections between existing nonlinear LRCs in finite fields and linear ring-based LRCs.