In case of an Abelian group \(A\) the authors investigate the problem of determining those rings \(R\) which are maximal as rings in the near-ring \(M_0(A)\) of all 0-preserving functions on \(A\). This problem is found to be related to properties of certain covers of \(A\) by subgroups. Special attention is given to the case where \(A\) is an elementary Abelian \(p\)-group. The authors define: \(C(R):=\{Ry\mid y\in A\) and \(Ry\) is maximal as a cyclic \(R\)-submodule of \(A\}\) and call this `the cover of \(A\) determined by \(R\)'; and note that \(C(R)\) is a (an irredundant) cover in that each cell, \(Ry\), contains an element in no other cell of the cover, or equivalently, no cell is contained in the union of the others. The word `cover' will mean an irredundant cover. A finite irredundant cover of a group \(A\) is a set \(\{C_i\}_{i=1}^s\) of subgroups of \(A\), called `cells', where \(A=\bigcup_{i=1}^sC_i\) and each \(C_i\) contains at least one element that is not in any other cell. Define \[ \Re(C):=\{\rho\in M_0(A)\mid\rho|_{C_i}\in\text{End}(C_i),\text{ for each cell }C_i\}, \] a subring (`the ring determined by \(C\)') of \(M_0 (A)\). A cover \(D=\{D_j\}\) is \(\Re(C)\)-invariant if \(\Re(C)D_j\subseteq D_j\) for each \(D_j\in D\), denoted \(C\leq D\) when \(C=\{C_i\}\) and \(D=\{D_i\}\) are covers of \(A\) and for each \(C_i\in C\), there exists some \(D_j\in D\) such that \(C_i\subseteq D_j\). For \(\Re(C)\subseteq\Hom(M,A)\), if \(M_1,\dots,M_t\) are maximal elements then \(\mathcal L\Re(C)\) means \(\{M_1,\dots M_t\}\). The main results established by the authors are: In case of a finite Abelian group \(A\), if \(R\) is a maximal subring of \(M_0 (A)\) then \(R=\Re(C)\) for some cover \(C\) of \(A\) and \(\Re(C)=\text{End}(A)\) if and only if \(C=\{A\}\). Also for a cover \(C\) of \(A\), \(\Re(C)\) is a maximal subring of \(M_0 (A)\) if and only if for each \(\Re(C)\)-invariant cover \(D\) of \(A\) with \(D\leq\mathcal L\Re(C)\), \(\Re(D)=\Re(C)\). Some other results are: \(R\) is a subring of \(M_0 (A)\). (i) With \(N_1\) and \(N_2\) as \(R\)-invariant subsets of \(A\) such that \(A=N_1\cup N_2\) and \(N_1\cap N_2=\{0\}\) and for \(i=1,2\), \(R\) contains the functions \(e_i\), that are the identity on \(N_i\) and zero off \(N_i\). Then \(R\cong R_1\oplus R_2\) where \(R_i\) is the ring \(R|_{N_i}\), \(i=1,2\) and (ii) If \(A\) has primary decomposition \(A=\bigoplus^t_{i=1}A_i\), \(|A_i|=p_i^{n_i}\) where \(p_i\) are distinct prime integers then \(R\cong R_1\oplus\cdots\oplus R_t\), where \(R_i\) is a subring of \(M_\mathbb{Z}(A_i)\subseteq M_0(A_i)\). Consequently, \(R\) is maximal in \(M_0(A)\) if and only if \(R_i\) is maximal in \(M_0(A_i)\), \(i=1,2,\dots,t\). Moreover when \(V\) is an elementary Abelian \(p\)-group the other result is: Let \(C\) be a cover of \(V\) with nontrivial cells \(C_1,\dots,C_k\). If \(k