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 $i\ne0, 1$.
  2. That's correct (despite the indices).

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