Missouri Journal of Mathematical Sciences

General Dorroh Extensions

I. Alhribat, P. Jara, and I. Márquez

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Abstract

In a recent paper G. A. Cannon and K. M. Neuerburg point out that if $A=\mathbb{Z}$ and $B$ is an arbitrary ring with unity, then $\mathbb{Z}\star{B}$, the Dorroh extension of $B$, is isomorphic to the direct product $\mathbb{Z}\times{B}$. Thus, the ideal structure of $\mathbb{Z}\star{B}$ can be completely described. The aim of this note is to point out that this result may be extended to any pair $(A,B)$ in which $B$ is an $A$-algebra with unity, and to study the construction of extensions of algebras without zero divisors and their behavior with respect to algebra maps.

Article information

Source
Missouri J. Math. Sci., Volume 27, Issue 1 (2015), 64-70.

Dates
First available in Project Euclid: 3 December 2015

Permanent link to this document
https://projecteuclid.org/euclid.mjms/1449161368

Digital Object Identifier
doi:10.35834/mjms/1449161368

Mathematical Reviews number (MathSciNet)
MR3431116

Zentralblatt MATH identifier
1337.16023

Subjects
Primary: 16D25: Ideals
Secondary: 16S70: Extensions of rings by ideals

Keywords
ring algebra Dorroh extension Szendrei extension

Citation

Alhribat, I.; Jara, P.; Márquez, I. General Dorroh Extensions. Missouri J. Math. Sci. 27 (2015), no. 1, 64--70. doi:10.35834/mjms/1449161368. https://projecteuclid.org/euclid.mjms/1449161368


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References

  • G. A. Cannon and K. M. Neuerburg, Ideals in Dorroh extensions of rings, Missouri J. Math. Sci., 20.3 (2008), 165–168.
  • T. de Alwis, The ideal structure of $\mathbb{Z}\star\mathbb{Z}$, Missouri J. Math. Sci., 6.2 (1994), 116–123.
  • J. L. Dorroh, Concerning adjunctions to algebras, Transactions of the Amer. Math. Soc., 26 (1932), 85–88.
  • J. Szendrei, On the extension of rings without divisors of zero, Acta. Univ. Szegred, 13 (1950), 231–234.