INSERTION-OF-FACTORS-PROPERTY WITH FACTORS MAXIMAL IDEALS Jin, Hai-Lan; Jung, Da Woon; Lee, Yang; Ryu, Sung Ju; Sung, Hyo Jin; Yun, Sang Jo;
Insertion-of-factors-property, which was introduced by Bell, has a role in the study of various sorts of zero-divisors in noncommutative rings. We in this note consider this property in the case that factors are restricted to maximal ideals. A ring is called IMIP when it satisfies such property. It is shown that the Dorroh extension of A by K is an IMIP ring if and only if A is an IFP ring without identity, where A is a nil algebra over a field K. The structure of an IMIP ring is studied in relation to various kinds of rings which have roles in noncommutative ring theory.
D. D. Anderson and V. Camillo, Armendariz rings and Gaussian rings, Comm. Algebra 26 (1998), no. 7, 2265-2272.
E. P. Armendariz, A note on extensions of Baer and P.P.-rings, J. Austral. Math. Soc. 18 (1974), 470-473.
H. E. Bell, Near-rings in which each element is a power of itself, Bull. Austral. Math. Soc. 2 (1970), 363-368.
S. Chhawchharia and M. B. Rege, Armendariz rings, Proc. Japan Acad. Ser. A Math. Sci. 73 (1997), no. 1, 14-17.
J. L. Dorroh, Concerning adjunctins to algebras, Bull. Amer. Math. Soc. 38 (1932), 85-88.
K. R. Goodearl, Von Neumann Regular Rings, Pitman, London, 1979.
J. Hannah and K. C. O'meara, Maximal quotient rings of prime group algebras, Proc. Amer. Math. Soc. 65 (1977), no. 1, 1-7.
C. Huh, S. H. Jang, C. O. Kim, and Y. Lee, Rings whose maximal one-sided ideals are two sided, Bull. Korean Math. Soc. 39 (2002), no. 3, 411-422.
C. Huh, Y. Lee, and A. Smoktunowicz, Armendariz rings and semicommutative rings, Comm. Algebra 30 (2002), no. 2, 751-761.
S. U. Hwang, Y. C. Jeon, and Y. Lee, Structure and topological conditions of NI rings, J. Algebra 302 (2006), no. 1, 186-199.
Y. C. Jeon, H. K. Kim, Y. Lee, and J. S. Yoon, On weak Armendariz rings, Bull. Korean Math. Soc. 46 (2009), no. 1, 135-146.
M. Kheradmand, H. K. Kim, T. K. Kwak, and Y. Lee, Reflexive property on nil ideals, Submitted.
N. K. Kim and Y. Lee, Armendariz rings and reduced rings, J. Algebra 223 (2000), no. 2, 477-488.
N. K. Kim and Y. Lee, Extensions of reversible rings, J. Pure Appl. Algebra 185 (2003), no. 1-3, 207-223.
L. Motais de Narbonne, Anneaux semi-commutatifs et unis riels anneaux dont les id aux principaux sont idempotents, Proceedings of the 106th National Congress of Learned Societies (Perpignan, 1981), 71-73, Bib. Nat., Paris, 1982.
G. Shin, Prime ideals and sheaf representation of a pseudo symmetric ring, Trans. Amer. Math. Soc. 184 (1973), 43-60.
A. Smoktunowicz, A simple nil ring exists, Comm. Algebra 30 (2002), no. 1, 27-59.