Insertion-of-factors-property on nilpotent elements. (Q2880479)

In this paper all rings are associative with identity unless otherwise stated. For a ring \(R\), \(N(R)\), \(N_*(R)\), \(N^*(R)\) denote the set of all nilpotent elements, the prime radical and the upper nilradical (that is, the sum of all nil ideals) of \(R\). A ring \(R\) is called an IFP ring if \(ab=0\) implies \(aRb=0\) for all \(a,b\in R\). A ring \(R\) (possibly without identity) is called NI if \(N^*(R)=N(R)\).NEWLINENEWLINE In this paper the authors introduce and study nil-IFP rings which form a generalization of NI rings. They call a ring \(R\) a `nil-IFP' ring if \(ab\in N(R)\) implies \(aRb\subseteq N(R)\). The authors show that the class of nil-IFP rings is closed under subrings (possibly without identity) but it is not closed under factor rings. They show that if \(I\) is a proper ideal of \(R\) and both \(R/I\) and \(I\) (as a ring without identity) are nil-IFP, then so is \(R\). Thus if \(I\) is a nil ideal of \(R\), then \(R\) is nil-IFP if and only if so is \(R/I\). Moreover, if \(e\) is a central idempotent of a ring \(R\), then \(R\) is nil-IFP if and only if \(eR\) and \((1-e)R\) are both nil-IFP.NEWLINENEWLINE A ring \(R\) is directly finite if \(ab=1\) implies \(ba=1\) for all \(a,b\in R\). The authors show that nil-IFP rings are directly finite but there exists a directly finite ring which is not nil-IFP.NEWLINENEWLINE A ring \(R\) is 2-primal if \(N_*(R)=N(R)\). The index of nilpotency of a nilpotent element \(b\) in a ring \(R\) is the least positive integer \(n\) such that \(b^n=0\). The index of nilpotency of a subset \(S\) of \(R\) is the supremum of the indices of nilpotency of all nilpotent elements in \(S\). If such a supremum is finite, then \(S\) is said to be of bounded index of nilpotency.NEWLINENEWLINE The authors show that for a ring \(R\) of bounded index of nilpotency, the following conditions are equivalent: (1) \(R\) is nil-IFP; (2) \(R\) is 2-primal; (3) \(R\) is NI. Thus every finite nil-IFP ring is 2-primal (hence NI). They show that if \(R\) is a nil-IFP ring but not NI, then there exists a countably infinite subring of \(R\) which is nil-IFP but not NI. The authors show that NI and nil-IFP properties are equivalent if and only if they are equivalent for countably infinite rings. Moreover, if Köthe's conjecture (that states that the upper nilradical contains all nil left ideals) holds, then every nil-IFP ring is NI. Also, the authors give some information about the set of nilpotent elements in nil-IFP rings which is too technical to specify in this review.NEWLINENEWLINE A ring is called Abelian if each idempotent is central. The authors show that nil-IFP rings need not be Abelian.NEWLINENEWLINE A ring \(R\) is called regular if for each \(a\in R\) there exists \(x\in R\) such that \(a=axa\). The authors prove that the following conditions are equivalent for a regular ring \(R\): (1) \(R\) is reduced (that is \(N(R)=0\)); (2) \(R\) is NI; (3) \(R\) is nil-IFP; (4) \(R\) is 2-primal; (5) \(R\) is Abelian; (6) \(R\) is IFP.NEWLINENEWLINE \(U_n(R)\) denotes the \(n\times n\) upper triangular matrix ring over a ring \(R\). The authors describe the class of minimal (that is, having the smallest cardinality) noncommutative nil-IFP rings. Namely, they show that if \(R\) is a minimal noncommutative nil-IFP ring with identity, then \(R\) is of order 8 and is isomorphic to \(U_2(\mathbb Z_2)\). But, if \(R\) is a minimal noncommutative nil-IFP ring without identity, then \(R\) is of order 4 and is isomorphic to \(R_i\) for some \(i\in\{1,2,3\}\), where \(R_1=\left(\begin{smallmatrix}\mathbb Z_2&\mathbb Z_2\\ 0&0\end{smallmatrix}\right)\), \(R_2=\left(\begin{smallmatrix} 0&\mathbb Z_2\\ 0&\mathbb Z_2\end{smallmatrix}\right)\) and \(R_3=\left(\begin{smallmatrix}\mathbb Z_2&0&0\\ 0&0&\mathbb Z_2\\ 0&0&0\end{smallmatrix}\right)\) are subrings of \(U_2(\mathbb Z_2)\) and \(U_3(\mathbb Z_2)\), respectively. Thus a ring \(R\) (possibly without identity) is a minimal noncommutative nil-IFP ring if and only if \(R\) is a minimal NI ring if and only if \(R\) is a minimal 2-primal ring.NEWLINENEWLINE In the final section of the paper, the authors examine the closure of the class of nil-IFP rings under some ring constructions. In particular, they show that: (1) a direct sum \(R=\sum_{i\in I}R_i\) of rings \(R_i\) is nil-IFP if and only if every \(R_i\) is nil-IFP; (2) the Dorroh extension \(D\) of an algebra \(R\) over a commutative ring \(S\) by \(S\) is nil-IFP if and only if so is \(R\); (3) for a ring \(R\) without identity, the ring \(R\times\mathbb Z\) (with the same operations as in the Dorroh extension) is nil-IFP if and only if so is \(R\); (4) the class of nil-IFP rings is closed under direct limits but it is not closed under direct products.NEWLINENEWLINE The \(n\times n\) lower triangular matrix ring over a ring \(R\) is denoted by \(L_n(R)\). The authors prove that the following conditions are equivalent for a ring \(R\) and an integer \(n\geqq 2\): (1) \(R\) is nil-IFP; (2) \(U_n(R)\) is nil-IFP; (3) \(L_n(R)\) is nil-IFP; (4) \(V_n(R)=\left\{\left(\begin{smallmatrix} a_1&a_2&a_3&\dots &a_{n-1}&a_n\\ 0&a_1&a_2&\dots&a_{n-2}&a_{n-1}\\ 0&0&a_1&\dots&a_{n-3}&a_{n-2}\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&0&\dots&a_1&a_2\\ 0&0&0&\dots&0&a_1\end{smallmatrix}\right):a_i\in R\right\}\) is nil-IFP, (5) \(R[x]/\langle x^n\rangle\) is a nil-IFP ring, where \(R[x]\) is the polynomial ring with an indeterminate \(x\) over \(R\) and \(\langle x^n\rangle\) is the ideal of \(R[x]\) generated by \(x^n\). The authors also show that if \(R\) and \(S\) are rings and \(_RM_S\) an \((R,S)\)-bimodule, then \(E=\left(\begin{smallmatrix} R&M\\ 0&S\end{smallmatrix}\right)\) is nil-IFP if and only if \(R\) and \(S\) are both nil-IFP.NEWLINENEWLINE The authors prove that the following conditions are equivalent for a ring \(R\) of bounded index of nilpotency: (1) \(R\) is nil-IFP; (2) \(R[x]\) is 2-primal; (3) \(R[x]\) is NI; (4) \(R[x]\) is nil-IFP.NEWLINENEWLINE The authors consider the nil-IFP condition of Ore extensions. For a ring \(R\), a ring endomorphism \(\sigma\colon R\to R\) and a \(\sigma\)-derivation \(\delta\colon R\to R\), the Ore extension \(R[x;\sigma,\delta]\) of \(R\) is the ring obtained by giving \(R[x]\) multiplication \(xr=\sigma(r)x+\delta(r)\) for all \(r\in R\). If \(\delta=0\), we write \(R[x;\sigma]\) for \(R[x;\sigma,0]\) and then the Ore extension is called a skew polynomial ring. If \(\sigma=1\), we write \(R[x;\delta]\) for \(R[x;1,\delta]\) and then the Ore extension is called a differential polynomial ring. The authors show that there exists a nil-IFP ring over which the skew polynomial ring is not nil-IFP. They also show that there exists a nil-IFP ring over which the differential polynomial ring is not nil-IFP.NEWLINENEWLINE A ring \(R\) is called nil-Armendariz if \(ab\in N(R)\) for every coefficient \(a\) of \(f(x)\) and every coefficient \(b\) of \(g(x)\) whenever \(f(x),g(x)\in R[x]\) satisfy \(f(x)g(x)\in N(R)[x]\). The authors close the paper by proving that the following conditions are equivalent for an Armendariz ring: (1) \(R\) is nil-IFP; (2) \(R\) is NI; (3) \(R\) is 2-primal; (4) \(R[x]\) is nil-IFP; (5) \(R[x]\) is NI; (6) \(R[x]\) is 2-primal. They also show that the condition Armendariz in this last result is not superfluous.