# Computing the homology of a simple chain complex

Let $$R$$ be a ring and $$x\in R$$ be a central element. Consider the complex

$$0 \rightarrow R \xrightarrow{x} R \rightarrow 0$$

concentrated in degrees 1 and 0. Compute the homology of this complex.

I have two questions:

1. What does it mean to say that this chain is concentrated in degrees 0 and 1?
2. Is my below attempt at homology correct?

So my guess is that the only non-trivial homologies are $$H_1 = Ker(x) = \{y \in R: xy = 0\}$$ and $$H_2 = R/(x)$$ where $$(x)$$ is the principal ideal generated by x. Can anything else be said here?

1. It means a complex $$\dots\to M_i\to\dots$$ such that $$M_i=0$$ for $$i\ne0, 1$$.