Reduced VS non-reduced cohomology ring

Question

Assume cohomology ring of topological space $X$ has the form:

$H^{*}(X) = \mathbb{Z}[x]/(x^k)$, where $x \in H^{1}(X)$ is a generator.

What can I say of its reduced cohomology ring $\tilde{H}^{*}(X)$? Will it just be free (non-unital) ring on generator $x$ modulo $x^k$ ?

Thanks.