# Integrals in noncommutative graded algebras which are not necessarily Hopf

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