Possible approaches to creating a magnetoresistive memory with recording by electric field (MERAM) are considered. The memory based on a particular geometry of multiferroic BiFeO3 is shown to be the most promising. The relatively small values of the weak ferromagnetic moment and linear magnetoelectric coupling constant in BiFeO3 are not significant obstructions for creation of such a MERAM. The key factors are the coupling between electrical polarization and the antiferromagnetism vector in the multiferroic (not between polarization and magnetization vectors), as well as the exchange coupling between the antiferromagnetism vector and the ferromagnet layer magnetization caused by the spin-flop orientation of the latter two vectors at the interface. This is inherently an antiferromagnetic-ferroelectric device mechanism.