Let X be a matrix of indeterminates, t an integer, and \(P_t(X)\) define the ideal generated by the permanents of all \(t\times t\) submatrix of X. \(P_t(X)\) is called a permanental ideal. In this article, we study the algebras \(\Bbbk [X]/P_t(X)\) where X is a generic, symmetric, or a Hankel matrix of indeterminates. When \({{\,\textrm{char}\,}}\Bbbk = 2\) , \(P_t(X)\) is also known as a determinantal ideal, a popular class in commutative algebra and algebraic geometry, and thus many properties of \(P_t(X)\) are known in this case. We prove that, if X is an \(n\times n\) matrix and \({{\,\textrm{char}\,}}\Bbbk >2\) , the algebra \(\Bbbk [X]/P_n(X)\) is F-regular, just like when \({{\,\textrm{char}\,}}\Bbbk = 2\) . On the other hand, we obtain a full characterization of when \(\Bbbk [X]/P_2(X)\) is F-pure or F-regular, when \({{\,\textrm{char}\,}}\Bbbk >2\) , and the answer is different than that in even characteristic.
