Missouri Journal of Mathematical Sciences

General Dorroh Extensions

I. Alhribat, P. Jara, and I. Márquez

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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

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

First available in Project Euclid: 3 December 2015

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

ring algebra Dorroh extension Szendrei extension


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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  • 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.