| Suppose R is a commutative integral domain, let Mmn(R) and Tmn(R) be themodule of m×n full matrices and the module of m×n upper triangular matrices overR, respectively. When m=n, Mmn(R) and Tmn(R) will be abbreviated to Mn(R)and Tn(R). By (?)(n1,…,nk) we designate the submodule of Mn(R) consisting of allblocked upper triangular matrices. Furthermore, for any i,j∈[1,n], Eij expressesthe matrix with 1 in the (i,j)th position and 0 elsewhere, the symbol N-1(X, Y)(orNt(X, Y)) denotes the set of f which is a preserving inverse linear map(or preservinggroup inverse linear map)from X to Y, X, Y∈{Tn(R), (?)(n1,…,nk)}.In this paper, we get rid of the restraint to R's characteristic, the linear surjec-tion in N-1(Tn(R)) are given directly when f(E1n)≠0. as the extend to module ofupper triangular matrices-module of blocked upper triangular matrices, we also getthe forms of the assemblage of N-1((?)(n1,…,nk), Mn(R)) when R is a principal idealdomain with 3 units at least and f is bijection. Furthermore, by restricting theintermediate matrix, the set of N-1((?)(n1,…,nk),(?)(n1,…,nk)) are characterized. As anapplication, the forms of the linear bijection in Tn(R) preserving inverse of matrixare also given.As the extend to the inverse of matrix-matrix's group inverse, when R is a prin-cipal ideal domain with 4 units at least, the linear bijection in Nt((?)(n1,…,nk), Mn(R))are described, by restricting the intermediate matrix, the set of Nt((?)(n1,…,nk), (?)(n1,…,nk))are characterized also. Meanwhile, the forms of the linear bijection in Tn(R) pre-serving group inverse of matrix are given too. |
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