 Square-zero element

An element of an associative algebra is square-zero if its square vanishes. Every such element is a nilpotent element. These elements are preserved by an algebra isomorphism, so their structure distinguishes algebras. Their set need not be a vector space: the matrix units $E_{12}$ and $E_{21}$ are square-zero, while their sum has square equal to the identity in $M_2(K)$. In a square-zero ideal, by contrast, every pairwise product vanishes and all its elements are square-zero. In $K[t]/(t^3)$ the square-zero elements are exactly the multiples of $t^2$, over every field: $(yt+z{t}^2{)}^2=y^2{t}^2$ forces $y=0$.