Homology of countable-dimensional real projective space

Over the integers
The homology groups with coefficients in the ring of integers $$\mathbb{Z}$$ are given as follows:

$$H_p(\mathbb{P}^\infty(\R);\mathbb{Z}) = \lbrace\begin{array}{rl}\mathbb{Z}/2\mathbb{Z}, &p = 1,3,5,\dots\\0, & p = 2,4,6, \dots \\ \mathbb{Z},& p = 0\\\end{array}$$

Over an abelian group or module $$M$$
The homology groups with coefficients in a module $$M$$ over a ring $$R$$ are given by:

$$H_p(\mathbb{P}^\infty(\R);M) = \lbrace\begin{array}{rl} M/2M, & p=1,3,5,\dots\\ T, & p = 2,4,6, \dots \\ M, & p = 0\\\end{array}$$

where $$T$$ is the 2-torsion submodule of $$M$$, i.e., the submodule of $$M$$ comprising elements whose double is zero.

In particular, we see the following cases:

Note that the third case, where $$M$$ is 2-divisible but not necessarily uniquely so, cannot arise if $$M = R$$ and it is a unital ring. So when taking coefficients over a unital ring, there is no need to distinguish between 2-divisibility and unique 2-divisibility.

Proof
The chain complex arising from a CW structure is as follows:

$$\dots \stackrel{\cdot 0}{\to} \mathbb{Z} \stackrel{\cdot 2}{\to} \mathbb{Z} \stackrel{\cdot 0}{\to} \mathbb{Z} \to \dots \stackrel{\cdot 2}{\to} \mathbb{Z} \stackrel{\cdot 0}{\to} \mathbb{Z}$$

where the subscript for the last written entry is $$0$$, and hence the multiplication by 2 maps arise from even to odd subscripts and the multiplication by zero maps arise from odd to even subscripts.

Homology computation over an abelian group or module $$M$$
The chain complex remains the same, but each $$\mathbb{Z}$$ is replaced by $$M$$.

Denote by $$T$$ the 2-torsion submodule of $$M$$ and by $$M/2M$$ the quotient of $$M$$ by the submodule $$2M$$ comprising the doubles of elements.