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Nanoring magnetic tunnel junction (NR-MTJs) with outer diameter of 90 nm, and nanoelliptical ring (NER) -shaped MTJ with major outer diameter of 120 nm, and minor outer diameter of 70 nm were fabricated. For both geometrical structures, the ring width was kept at around 30 nm. The magnetic field driven and current-induced magnetization switching of the NR-MTJs and NER-MTJs were investigated. Compared with the NR-MTJs, the NER-MTJs show a sharp magnetization switching and lower coercivity under the sweep of an external magnetic field, which indicates the higher uniaxial anisotropy of NER-MTJs. In addition, the current-induced magnetization switching was achieved in both MTJs which indicated a lower critical current density in sandwiched-type and spin-valve-type NER-MTJs. This reduction of critical current could be resulting from different distribution of current-induced Oersted field in the NR-MTJs and NER-MTJs, which play an assisted role in spin transfer torque switching. This work proposes the NER-MTJs with the low current density, and small stray field, as data storage bits of magnetic random access memory (MRAM), could be a promising candidate for further reducing the power consumption, scaling memory cell size and increasing density of MRAM.