TY  - JOUR
T1  - On a property of the ideals of the polynomial ring $R[x]$
AU  - Al-maktry, Amr Ali Abdulkader
PY  - 2022
DA  - January
DO  - 10.24330/ieja.1058380
JF  - International Electronic Journal of Algebra
JO  - IEJA
PB  - Abdullah HARMANCI
WT  - DergiPark
SN  - 1306-6048
SP  - 1
EP  - 12
VL  - 31
IS  - 31
LA  - en
AB  - Let RR be a commutative ring with unity 1≠01≠0. In this paper we introduce  the definition of the first derivative property on the ideals of  the polynomial ring R[x]R[x]. In particular,   when RR is a finite local ring with principal maximal ideal m≠{0}m≠{0}  of  index of nilpotency ee, where 1<e≤|R/m|+11≤e≤|R/m|+1,  we show that the null ideal consisting of polynomials inducing the zero function on RR  satisfies this property. As an application,  when RR  is a finite local ring with  null ideal satisfying this property,  we  prove  that the stabilizer group of RR in the group of   polynomial permutations on the ring R[x]/(x2)R[x]/(x2),  is isomorphic  to a certain factor group of the null ideal.
KW  - Commutative rings
KW  - polynomial ring
KW  - null ideal
KW  - null polynomial
KW  - Henselian ring
KW  - finite local ring
KW  - dual numbers
KW  - polynomial permutation
KW  - permutation
polynomial
KW  - finite permutation group
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CR  - M. W. Rogers and C. Wickham, Polynomials inducing the zero function on local rings, Int. Electron. J. Algebra, 22 (2017), 170-186.
UR  - https://doi.org/10.24330/ieja.1058380
L1  - https://dergipark.org.tr/en/download/article-file/2194334
ER  -