Checking codes play a role in ensuring credibility in delivering information in the noisy communication channels. The paper is a detailed ring-theoretic analysis of error-correcting codes, in the algebraic structure of the linear, cyclic and constacyclic codes of finite rings. In certain applications, the ring-based coding structures are more flexible, have better structural richness and better error-handling capabilities than classical field-based approaches. Algebraic definitions of linear codes (as modules), the construction of cyclic and constacyclic codes in commutative rings, and the geometry of such codes (duality, minimal generating sets and ideal decomposition) are discussed in the paper. In addition, the efficient decoding algorithms which are unique to the ring-based codes are discussed with a focus on their efficiency in computation, and their use in practice. The results indicate that the codes using the ring-theoretic methods are far superior in code performance and flexibility and can thus be used in the existing communication system, e.g., storage, wireless resource and cryptographic protocols.