Let $\mathbf{k}$ be a field. Let $A$ be a finite dimensional $\mathbb{Z}_{\geq 0}$-graded $\mathbf{k}$-algebra such that $A^0=\mathbf{k}1$. Let $m$ be the maximal non-negative integer such that $A^m\neq 0$. Every element $P\in A^m\setminus\{0\}$ satisfies $AP=PA=\mathbf{k}P\neq 0$, i.e. $P$ generates a one-dimensional two-sided ideal in $A$.
On the other hand, suppose that $I$ is a one-dimensional two-sided ideal in $A$. Under what hypotheses can we conclude that $I\subseteq A^m$?
One example of such hypotheses would be that $A$ is in addition a Hopf algebra or braided Hopf algebra by the theory of integrals ([1, Corollary 2.7, Equivalence (2.11)$\Leftrightarrow$(2.13)] and [2, Subsection 2.3]). Do you know other hypotheses without the Hopf assumption?
Remark. The hypotheses should have nothing to do with commutativity. The algebra $A$ has to be considered as noncommutative.
[1] M. E. Sweedler, Integrals for Hopf algebras, Ann. of Math. (2) 89 (1969), 323-335. MR 0242840
[2] N. Andruskiewitsch and M. Graña, Braided Hopf algebras over non-abelian finite groups, Bol. Acad. Nac. Cienc. (Córdoba) 63 (1999), 45-78, Colloquium on Operator Algebras and Quantum Groups (Spanish) (Vaquerías, 1997). MR 17145450 arXiv:math/9802074
