# ALGEBRAS WITH A NILPOTENT GENERATOR OVER ℤp2

• Woo, Sung-Sik (Department of Mathematics, Ewha Women's University)
• Published : 2006.08.01
• 95 6

#### Abstract

The purpose of this paper is to describe the structure of the rings $\mathbb{Z}_{p^2}[X]/({\alpha}(X))$ with ${\alpha}(X)$ a monic polynomial and $\={X}^{\kappa}=0$ for some nonnegative integer ${\kappa}$. Especially we will see that any ideal of such rings can be generated by at most two elements of the special form and we will find the 'minimal' set of generators of the ideals. We indicate how to identify the isomorphism types of the ideals as $\mathbb{Z}_{p^2}-modules$ by finding the isomorphism types of the ideals of some particular ring. Also we will find the annihilators of the ideals by finding the most 'economical' way of annihilating the generators of the ideal.

#### Keywords

cyclic code over $\mathbb{Z}_4$

#### References

1. M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley, 1969
2. P. Kanwar and S. R. Lopez-Permouth, Cyclic codes over the integer modulo $p^n$, Finite fields Appl. 3 (1997), no. 2, 334-352 https://doi.org/10.1006/ffta.1997.0189
3. S. S. Woo, Cyclic codes of length $2^n$ over $Z_4$, preprint, 2004

#### Cited by

1. IDEALS OF Zpn[X]/(Xl-1) vol.26, pp.3, 2011, https://doi.org/10.4134/CKMS.2011.26.3.427