ON A SPECIAL PRESENTATION OF MATRIX ALGEBRAS

Geir AGNARSSON; Samuel S. MENDELSON

doi:https://doi.org/10.24330/ieja.662946

Makale detayları

ON A SPECIAL PRESENTATION OF MATRIX ALGEBRAS

Yazarlar:

Geir AGNARSSON | Samuel S. MENDELSON

Yayıncı:

Abdullah HARMANCI

Ana Başlıklar:

Matematik

Dergi:

International Electronic Journal of Algebra 2020 Vol.27 Issue 27, pp.13-42

Belge Tipi:

Makale

Yayın Tarihi:

1/2020

Belge Dili:

İngilizce

Document ID (Org No.):

2000717895

Sıklık:

Aylık

eISSN:

1306-6048

pISSN:

1306-6048

DOI:

https://doi.org/10.24330/ieja.662946

Özet:

 

Recognizing when a ring is a complete matrix ring is of significant importance in algebra.  It is well-known folklore that a ring RRR is a complete n×nn×nn\times n matrix ring, so R≅Mn(S)R≅Mn(S)R\cong M_{n}(S) for some ring SSS, if and only if it contains a set of n×nn×nn\times n matrix units {eij}ni,j=1{eij}i,j=1n\{e_{ij}\}_{i,j=1}^n. A more recent and less known result states that a ring RRR is a complete (m+n)×(m+n)(m+n)×(m+n)(m+n)\times(m+n) matrix ring if and only if, RRR contains three elements, aaa, bbb, and fff, satisfying the two relations afm+fnb=1afm+fnb=1af^m+f^nb=1 and fm+n=0fm+n=0f^{m+n}=0. In many instances the two elements aaa and bbb can be replaced by appropriate powers aiaia^i and ajaja^j of a single element aaa respectively. In general very little is known about the structure of the ring SSS. In this article we study in depth the case m=n=1m=n=1m=n=1 when R≅M2(S)R≅M2(S)R\cong M_2(S). More specifically we study the universal algebra over a commutative ring AAA with elements xxx and yyy that satisfy the relations xiy+yxj=1xiy+yxj=1x^iy+yx^j=1 and y2=0y2=0y^2=0. We describe completely the structure of these AAA-algebras and their underlying rings when gcd(i,j)=1gcd(i,j)=1\gcd(i,j)=1. Finally we obtain results that fully determine when there are surjections onto M2(\field)M2(\field)M_2(\field) when \field\field\field is a base field \rats\rats\rats or \intsp\intsp\ints_p for a prime number ppp.

Makale detayları

ON A SPECIAL PRESENTATION OF MATRIX ALGEBRAS

Yazarlar: Geir AGNARSSON | Samuel S. MENDELSON

Yayıncı: Abdullah HARMANCI

Ana Başlıklar: Matematik

Dergi: International Electronic Journal of Algebra 2020 Vol.27 Issue 27, pp.13-42

Belge Tipi: Makale

Yayın Tarihi: 1/2020

Belge Dili: İngilizce

Document ID (Org No.): 2000717895

Sıklık: Aylık

eISSN: 1306-6048

pISSN: 1306-6048

DOI: https://doi.org/10.24330/ieja.662946

Sayı

Yazarlar:

Geir AGNARSSON | Samuel S. MENDELSON

Yayıncı:

Abdullah HARMANCI

Ana Başlıklar:

Matematik

Dergi:

International Electronic Journal of Algebra 2020 Vol.27 Issue 27, pp.13-42

Belge Tipi:

Makale

Yayın Tarihi:

1/2020

Belge Dili:

İngilizce

Document ID (Org No.):

2000717895

Sıklık:

Aylık

eISSN:

1306-6048

pISSN:

1306-6048

DOI:

https://doi.org/10.24330/ieja.662946