Certain Maximal Commutative Subrings of Full Matrix Rings

Ruangvarin Intarawong Sararnrakskul; Patcharee Lertvijitsilp; Amorn Wasanawichit; Sajee Pianskool

Authors

Ruangvarin Intarawong Sararnrakskul
Patcharee Lertvijitsilp
Amorn Wasanawichit
Sajee Pianskool

Keywords:

Full matrix ring, maximal commutative subring.

Abstract

Denote by $gif.latex?M_{n}(R)$ the full matrix ring over a commutative ring $gif.latex?R$ with identity where $gif.latex?n$ > $gif.latex?1$ . In this paper, we show that the set $gif.latex?D_{n}(R)$ of all matrices in $gif.latex?M_{n}(R)$ of the form

$gif.latex?\left[\begin{array}{ccccc}&space;x_{1}&space;&&space;0&space;&&space;\cdots&space;&&space;0&space;&&space;y_{1}\\&space;0&space;&&space;x_{2}&space;&&space;\cdots&space;&&space;y_{2}&space;&&space;0\\&space;\cdots&space;&&space;\cdots&space;&&space;\cdots&space;&&space;\cdots&space;&&space;\cdots&space;\\&space;0&space;&&space;y_{2}&space;&&space;\cdots&space;&&space;x_{2}&space;&&space;0\\&space;y_{1}&space;&&space;0&space;&&space;\cdots&space;&&space;0&space;&&space;x_{1}\\&space;\end{array}\right&space;]$

is a maximal commutative subring of the ring $gif.latex?M_{n}(R)$ .

Certain Maximal Commutative Subrings of Full Matrix Rings

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