﻿ 域上矩阵保乘积的诱导映射 Induced Maps Preserving Multiplicative Matrices over Fields

Pure Mathematics
Vol.06 No.03(2016), Article ID:17522,6 pages
10.12677/PM.2016.63025

Induced Maps Preserving Multiplicative Matrices over Fields

Jun Zhang, Chongguang Cao

Heilongjiang University, Harbin Heilongjiang

Received: Apr. 25th, 2016; accepted: May 9th, 2016; published: May 12th, 2016

ABSTRACT

Let F be a field, be the set of all matrices over F. If a map is defined by

where is the set of functions on F, then f is called a map induced by on. If implies, then f is called preserving multiplicative matrices. In this paper, we characterize induced maps preserving multiplicative matrices over fields.

Keywords:Field, Preserving Multiplicative, Induced Map

1. 引言

2. 主要结果

(1)

。 (2)

1) 对于任意不同的及任意均有。由此情形条件可知

， (3)

。 (4)

，易见

， (5)

， (6)

， (7)

，易见

(8)

， (9)

，由(7)可知，当时，有

， (10)

，又由(10)得

，则由(7)计算可得，则有

， (11)

， (12)

， (13)

，则中，时，有，则

， (14)

2) 若存在某及某使

，由，看

， (15)

， (16)

，看位置，得

， (17)

。 (18)

1) 对于任意不同的及任意均有

， (19)

， (20)

， (21)

， (22)

，令

。 (23)

2) 对于某

， (24)

， (25)

， (26)

，由，看(1,1)位置，又由(4)得

， (27)

， (28)

， (29)

Induced Maps Preserving Multiplicative Matrices over Fields[J]. 理论数学, 2016, 06(03): 166-171. http://dx.doi.org/10.12677/PM.2016.63025

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