• Journal of the Korean Mathematical Society
• /
• v.53 no.2
• /
• pp.415-431
• /
• 2016

This note is concerned with examining nilradicals and Jacobson radicals of polynomial rings when related factor rings are Armendariz. Especially we elaborate upon a well-known structural property of Armendariz rings, bringing into focus the Armendariz property of factor rings by Jacobson radicals. We show that J(R[x]) = J(R)[x] if and only if J(R) is nil when a given ring R is Armendariz, where J(A) means the Jacobson radical of a ring A. A ring will be called feckly Armendariz if the factor ring by the Jacobson radical is an Armendariz ring. It is shown that the polynomial ring over an Armendariz ring is feckly Armendariz, in spite of Armendariz rings being not feckly Armendariz in general. It is also shown that the feckly Armendariz property does not go up to polynomial rings.

• Communications of the Korean Mathematical Society
• /
• v.34 no.1
• /
• pp.127-136
• /
• 2019

This article concerns the normal property of elements on Jacobson and nil radicals which are generalizations of commutativity. A ring is said to be right njr if it satisfies the normal property on the Jacobson radical. Similarly a ring is said to be right nunr (resp., right nlnr) if it satisfies the normal property on the upper (resp., lower) nilradical. We investigate the relations between right duo property and the normality on Jacobson (nil) radicals. Related examples are investigated in the procedure of studying the structures of right njr, nunr, and nlnr rings.

• Journal of the Korean Mathematical Society
• /
• v.38 no.6
• /
• pp.1117-1123
• /
• 2001

A ring is a DICC ring if every chain of right ideals in-dexed by the integers stabilizes to the left or to the right or to both sides. A counterexample is given to an assertion of karamzadeh and Motamedi that a transfinite power of the Jacobson radical of a right DICC ring is zero. we determine the behavior of the transfinite powers of the Jacobson radical relative to a torsion theory and consequently can obtain their correct behavior in the classical setting.

• Bulletin of the Korean Mathematical Society
• /
• v.38 no.1
• /
• pp.121-128
• /
• 2001

In this paper, we define Jacobson modules which are the generalization of Jacobson rings. We give criteria of Jacobson modules and useful properties of Jacobson modules.

• East Asian mathematical journal
• /
• v.34 no.1
• /
• pp.39-46
• /
• 2018

In this article we consider rings whose Jacobson radical contains all the nilpotent elements, and call such a ring an NJ-ring. The class of NJ-rings contains NI-rings and one-sided quasi-duo rings. We also prove that the Koethe conjecture holds if and only if the polynomial ring R[x] is NJ for every NI-ring R.

• Korean Journal of Mathematics
• /
• v.18 no.1
• /
• pp.97-103
• /
• 2010

It is well known that the group ring K[G] has the nontrivial Jacobson radical if K is a field of characteristic p and G is a finite group of which order is divided by a prime p. This paper is concerned with the structure of the Jacobson radical of such a group ring.

• Journal of the Korean Mathematical Society
• /
• v.55 no.5
• /
• pp.1193-1205
• /
• 2018

We focus our attention on a ring property that nilpotents are contained in the Jacobson radical. This property is satisfied by NI and left (right) quasi-duo rings. A ring is said to be NJ if it satisfies such property. We prove the following: (i) $K{\ddot{o}}the^{\prime}s$ conjecture holds if and only if the polynomial ring over an NI ring is NJ; (ii) If R is an NJ ring, then R is exchange if and only if it is clean; and (iii) A ring R is NJ if and only if so is every (one-sided) corner ring of R.

• Bulletin of the Korean Mathematical Society
• /
• v.41 no.4
• /
• pp.599-608
• /
• 2004

Let R be an associative ring with identity and J(R) be the Jacobson radical of R. In this paper we investigate the generalization of the Jacobson radical of R, J＊ (R) say. Also we study the rings that J＊(R) = J(R).

• Communications of the Korean Mathematical Society
• /
• v.15 no.3
• /
• pp.453-467
• /
• 2000

For any S-flat module RM(which will be called endoflat) with a commutaitve ring R with identity, where S is the endomorphism ring RM, the fact that every epimorphism is an automorphism has been proved and the Jacobson Radical Rad(S) of S is described as follow; Rad(S) = { f$\in$S｜Imf=Mf is small in M} = {f$\in$S｜Imf $\leq$Rad(M)}. Additionally for any quasi-injective endo-flat module RM, the fact that every monomorphism is an automorphism has been proved and the Jacobson Radical Rad(S) for any quasi-injective endo-flat module has been studied too. Also some equivalent conditions for the semi-primitivity of any faithful endo-flat module RM with the open Jacobson Radical Rad(M) and those for the semi-simplicity of any faithful endo-flat quasi-injective module RM with the closed Socle Soc(M) have been studied.

• Kyungpook Mathematical Journal
• /
• v.54 no.4
• /
• pp.595-606
• /
• 2014

In this paper three more right Jacobson-type radicals, $J^r_{g{\nu}}$, are introduced for near-rings which generalize the Jacobson radical of rings, ${\nu}{\in}\{0,1,2\}$. It is proved that $J^r_{g{\nu}}$ is a special radical in the class of all near-rings. Unlike the known right Jacobson semisimple near-rings, a $J^r_{g{\nu}}$-semisimple near-ring R with DCC on right ideals is a direct sum of minimal right ideals which are right R-groups of type-$g_{\nu}$, ${\nu}{\in}\{0,1,2\}$. Moreover, a finite right $g_2$-primitive near-ring R with eRe a non-ring is a near-ring of matrices over a near-field (which is isomorphic to eRe), where e is a right $g_2$-primitive idempotent in R.