The row rank of a subring of a matrix ring

Authors:
M. S. Li and J. M. Zelmanowitz

Journal:
Proc. Amer. Math. Soc. **99** (1987), 627-633

MSC:
Primary 16A42; Secondary 15A30

MathSciNet review:
877029

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Abstract: For a subring of an matrix ring over a division ring , we examine an invariant called the row rank of . Roughly speaking, the row rank of is the largest integer such that contains all -rowed matrices over a left order in . The row rank of is then an integer between 0 and ; and we will see that row rank means that is the maximal left quotient ring of , while row rank signifies that is the classical left quotient ring of . Thus row rank provides a link between maximal and classical quotient rings for rings of this type. A description of the subrings with row rank is obtained which subsumes and generalizes earlier theorems of Faith-Utumi and Zelmanowitz, respectively, for the cases row rank and row rank .

**[1]**Carl Faith,*Lectures on injective modules and quotient rings*, Lecture Notes in Mathematics, No. 49, Springer-Verlag, Berlin-New York, 1967. MR**0227206****[2]**Carl Faith and Yuzo Utumi,*On Noetherian prime rings*, Trans. Amer. Math. Soc.**114**(1965), 53–60. MR**0172895**, 10.1090/S0002-9947-1965-0172895-4**[3]**Nathan Jacobson,*Structure of rings*, American Mathematical Society Colloquium Publications, Vol. 37. Revised edition, American Mathematical Society, Providence, R.I., 1964. MR**0222106****[4]**J. M. Zelmanowitz,*Large rings of matrices contain full rows*, J. Algebra**73**(1981), no. 2, 344–349. MR**640041**, 10.1016/0021-8693(81)90326-4**[5]**-,*On the rank of subrings of matrix rings*, Proc. 4th Internat. Conf. on Representations of Algebras (V. Dlab, Ed.), Carleton Univ., Ottawa, 1984.

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DOI:
https://doi.org/10.1090/S0002-9939-1987-0877029-6

Article copyright:
© Copyright 1987
American Mathematical Society