## Missouri Journal of Mathematical Sciences

### General Dorroh Extensions

#### 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

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

#### 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

#### 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.