投射维度、内射维度与同调维度（又称整体维度）是交换代数中考虑的重要不变量。

以下设 为交换环，而 为 -模。

的内射维度 定义为其内射分解的最短长度（当 时置 ）。投射维度 则定义为其投射分解的最短长度。

利用同调代数的工具，可以进一步得到下述刻划：

命题一. 设 为整数，下述条件等价：

命题二. 设 为整数，下述条件等价：

当 为诺特环而 为有限生成 -模时，上述条件更等价于

由此可定义环 的同调维度 为：

In ring theory and homological algebra, the global dimension (or global homological dimension; sometimes just called homological dimension) of a ring A denoted gl dim A, is a non-negative integer or infinity which is a homological invariant of the ring. It is defined to be the supremum of the set of projective dimensions of all A-modules. Global dimension is an important technical notion in the dimension theory of Noetherian rings. By a theorem of Jean-Pierre Serre, global dimension can be used to characterize within the class of commutative Noetherian local rings those rings which are regular. Their global dimension coincides with the Krull dimension, whose definition is module-theoretic.