Zero divisors in an algebra with two generators

Question

Let $k$ be a field, and $R = k\langle x,y \mid x^2 = 0\rangle$. The elements $x$ and $y$ are not supposed to commute with each other. Is the only case where nonzero elements $a, b \in R$ satisfy $ab=0$ when $a\in Rx$ and $b\in xR$?

Answer 1

The question is answered in Example 9.3 in the paper Victor Camillo and Pace P. Nielsen, McCoy rings and zero-divisors, J. Pure Appl. Algebra 212 (2008), no. 3, 599–615.