This paper introduces the concept of relative homological algebra, focusing on functors analogous to "Tor" and "Ext" from Cartan-Eilenberg's work but applied to module theory relative to a given subring. The authors aim to show how certain relative cohomology theories for groups, rings, and Lie algebras can be subsumed under the theory of the relative Ext functor. They define $(R, S)$-injective and $(R, S)$-projective modules and construct the relative Tor and Ext functors. The paper discusses the properties of these functors, including their behavior under exact sequences and their applications in extending cohomology theories. Specifically, it explores how the relative Ext functor can be used to extend cohomology theories for algebras, groups, and Lie algebras. The paper also provides explicit $(R, S)$-projective resolutions for certain modules and derives theorems that relate relative cohomological dimension to relative global ring dimension. Finally, it generalizes results from Cartan-Eilenberg to show that for a commutative ring $Q$ and the polynomial ring $P = Q[x_1, \cdots, x_n]$, the relative global dimension and relative cohomological dimension of $(P, Q)$ are equal to $n$.This paper introduces the concept of relative homological algebra, focusing on functors analogous to "Tor" and "Ext" from Cartan-Eilenberg's work but applied to module theory relative to a given subring. The authors aim to show how certain relative cohomology theories for groups, rings, and Lie algebras can be subsumed under the theory of the relative Ext functor. They define $(R, S)$-injective and $(R, S)$-projective modules and construct the relative Tor and Ext functors. The paper discusses the properties of these functors, including their behavior under exact sequences and their applications in extending cohomology theories. Specifically, it explores how the relative Ext functor can be used to extend cohomology theories for algebras, groups, and Lie algebras. The paper also provides explicit $(R, S)$-projective resolutions for certain modules and derives theorems that relate relative cohomological dimension to relative global ring dimension. Finally, it generalizes results from Cartan-Eilenberg to show that for a commutative ring $Q$ and the polynomial ring $P = Q[x_1, \cdots, x_n]$, the relative global dimension and relative cohomological dimension of $(P, Q)$ are equal to $n$.