Mutually unbiased maximally entangled bases in $$mathbb {C}^dotimes mathbb {C}^d$$

We study mutually unbiased maximally entangled bases (MUMEB’s) in bipartite system (mathbb {C}^dotimes mathbb {C}^d (d ge 3)). We generalize the method to construct MUMEB’s given in Tao et al. (Quantum Inf Process 14:2291–2300, 2015), by using any commutative ring R with d elements and generic character of ((R,+)) instead of (mathbb {Z}_d=mathbb {Z}/dmathbb {Z}). Particularly, if (d=p_1^{a_1}p_2^{a_2}ldots p_s^{a_s}) where (p_1, ldots , p_s) are distinct primes and (3le p_1^{a_1}le cdots le p_s^{a_s}), we present (p_1^{a_1}-1) MUMEB’s in (mathbb {C}^dotimes mathbb {C}^d) by taking (R=mathbb {F}_{p_1^{a_1}}oplus cdots oplus mathbb {F}_{p_s^{a_s}}), direct sum of finite fields (Theorem 3.3).