Open AccessOn irreducibility of monomial 7 × 7−matrix over local ring
Author(s) -
Alexander Tylyshchak
Publication year - 2018
Publication title -
vìsnik. serìâ fìziko-matematičnì nauki/vìsnik kiì̈vsʹkogo nacìonalʹnogo unìversitetu ìmenì tarasa ševčenka. serìâ fìziko-matematičnì nauki
Language(s) - English
Resource type - Journals
eISSN - 2218-2055
pISSN - 1812-5409
DOI - 10.17721/1812-5409.2018/3.5
Subject(s) - mathematics , matrix ring , jacobson radical , irreducibility , ring (chemistry) , monomial , combinatorics , matrix (chemical analysis) , commutative ring , identity matrix , pure mathematics , zero (linguistics) , discrete mathematics , commutative property , invertible matrix , chemistry , organic chemistry , linguistics , materials science , eigenvalues and eigenvectors , physics , philosophy , composite material , quantum mechanics
We consider a monomial n × n-matrix, which corresponds to a cyclic permutation of the length n, over a commutative local principle ideals ring. Non-zero elements of a non-empty set of first columns of the matrix are identity element of the ring and non-zero elements of non-empty set of the rest columns are a fixed non-zero generator element of the Jacobson radical of the ring. It is known if number of identities or number of generator elements is exact 1 or if n < 7 and number of identities is relatively prime to n, then the matrix is irreducible. If the number of identities is not relatively prime to n, then the matrix is reducible. If the Jacobson radical of the ring is nilpotent of degree 2, then the 7 × 7-matrix of considered form with 3 or 4 identities is reducible. It has been shown that the 7 × 7-matrix is irreducible if the degree of nilpotency of the Jacobson radical of the ring is higher than 2. Some necessary conditions of reducibility of this square matrix of arbitrary size are also established.