Bortosh M. Linear-algebraic properties of categories of monomial matrices over local rings

The thesis is devoted to the study of matrices over commutative rings. It is proved that a canonically cyclic matrix (over a commutative local ring of principal ideals) with nonperiodic weight sequences is indecomposable. Necessary and sufficient conditions (separately) for the indecomposability of a canonically cyclic matrix with a periodic weight sequence are indicated. It is proved that canonically cyclic matrices are isomorphic in the category of monomial matrices if and only if they are isomorphic in the strong category of monomial matrices. We prove a necessary condition for the irreducibility of a canonically t-cyclic matrix of arbitrary weight, in which a relation between the order and the weight of the matrix plays a major role. This condition is generalized to a certain class canonically (t; *)-cyclic matrices (in both cases, the commutative local ring is not necessarily a ring of principal ideals). We obtain a criterion for the irreducibility of a canonically t-cyclic matrix over a ring with unique decomposition and the criterion for the hereditary irreducibility of a canonically t-cyclic matrix of small weight over a commutative ring of principal ideals. These criteria are generalized to a certain class of canonically (t; *)-cyclic matrices. We study necessary conditions of irreducibility and sufficient conditions of reducibility of canonically cyclic matrices of larger weight over a commutative local ring. These conditions are considered with respect to the sequences of the special form of the weight sequence. In the study of canonical (t; *)-cyclic matrices we consider variuos types of the reducibilities: (*; 2)-reducibility, (*; 3)-reducibility, 2-hereditary reducibility.