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Applying the classical work of Nakayama [Ann. of Math. 40 (1939)], we exhibit a general form of non-Frobenius self-injective finite-dimensional algebras over a field. 2000 Mathematics Subject Classification: 16D50, 16L60.doi:10.4064/cm105-1-12 fatcat:fcu4wgezizh2ncicsiwfvqplzu