Since the theory of filters plays an important role in the theory of lattices, in this paper, we will make an intensive study of the notions of semisimple lattices and the socle of lattices based on their filters. The bulk of this paper is devoted to stating and proving analogues to several well-known theorems in the theory of the rings. It is shown that, if $L$ is a semisimple distributive lattice, then $L$ is finite. Also, an application of the results of this paper is given. It is shown that if $R$ is a right distributive ring, then the lattice of right ideals of $R$ is semisimple iff $R$ is a semisimple ring.
Ebrahimi Atani, S., Khoramdel, M., Dolati Pish Hesari, S., & Nikmard Rostam Alipour, M. (2022). Semisimple lattices with respect to filter theory. Journal of Algebra and Related Topics, 10(2), 131-143. doi: 10.22124/jart.2022.21065.1342
MLA
S. Ebrahimi Atani; M. Khoramdel; S. Dolati Pish Hesari; M. Nikmard Rostam Alipour. "Semisimple lattices with respect to filter theory". Journal of Algebra and Related Topics, 10, 2, 2022, 131-143. doi: 10.22124/jart.2022.21065.1342
HARVARD
Ebrahimi Atani, S., Khoramdel, M., Dolati Pish Hesari, S., Nikmard Rostam Alipour, M. (2022). 'Semisimple lattices with respect to filter theory', Journal of Algebra and Related Topics, 10(2), pp. 131-143. doi: 10.22124/jart.2022.21065.1342
VANCOUVER
Ebrahimi Atani, S., Khoramdel, M., Dolati Pish Hesari, S., Nikmard Rostam Alipour, M. Semisimple lattices with respect to filter theory. Journal of Algebra and Related Topics, 2022; 10(2): 131-143. doi: 10.22124/jart.2022.21065.1342
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