Accessible subrings and Kurosh's chains of associative rings. (Q2873588)

This paper concerns associative rings only. As usual, \(I\vartriangleleft R\) (\(I<R\), \(I<_lR)\) means that \(I\) is an ideal (one-sided ideal, left ideal) of a ring \(R\). A subring \(A\) of a ring \(R\) is said to be \(n\)-accessible (one-sided \(n\)-accessible, left \(n\)-accessible) in \(R\) if there are subrings \(R=A_0,A_1,\ldots,A_n=A\) of \(R\) such that \(A_i\vartriangleleft A_{i-1}\) (\(A_i<A_{i-1}\), \(A_i<_lA_{i-1}\)) for \(i=1,2,\ldots,n\). \(A\) is called precisely \(n\)-accessible (precisely one-sided \(n\)-accessible, precisely left \(n\)-accessible) if it is \(n\)-accessible (one-sided \(n\)-accessible, left \(n\)-accessible) in \(R\) but not \(k\)-accessible, (one-sided \(k\)-accessible, left \(k\)-accessible) in \(R\) for any positive integer \(k<n\). A subring \(A\) is accessible (one-sided accessible, left accessible) in \(R\) if it is \(n\)-accessible (one-sided \(n\)-accessible, left \(n\)-accessible) in \(R\) for some natural number \(n\). A ring \(R\) is filial if every accessible subring of \(R\) is an ideal of \(R\). A subring \(A\) of a ring \(R\) is essential in \(R\) if \(A\cap I\neq 0\) for every nonzero ideal \(I\) of \(R\). A ring \(R\) is an iterated maximal essential extension of \(A\) (written \(R=\mathrm{IME}(A)\)) if \(A\) is an essential accessible subring of \(R\) such that for every ring \(S\) in which \(A\) is accessible, there exists a homomorphism of \(S\) into \(R\) which is the identity map on \(A\). For a given ring \(R\) and \(a\in R\), \([a]\) denotes the subring of \(R\) generated by \(a\). Let \(R\) be an integral domain with the field of quotients \(K\). \(R\) is called a completely normal ring if for any \(x\in K\) and \(0\neq a\in R\), we have that \([x]a\subseteq R\) implies \(x\in R\).NEWLINENEWLINE A radical class \(S\) is called stable (left stable) if for every \(L<R\) (\(L<_lR)\) the radical \(S(L)\) of \(L\) is contained in the radical \(S(R)\) of \(R\). The smallest left stable radical containing a homomorphically closed class \(\mathcal N\) is denoted by \(st(\mathcal N)\).NEWLINENEWLINE Let \(\mathcal M\) be a homomorphically closed class of associative rings. Let \(\mathcal M_1=\mathcal M^1=\mathcal M_l^1=\mathcal M\) and for ordinals \(\alpha\) greater or equal than \(2\), define \(\mathcal M_\alpha\) (\(\mathcal M^\alpha\), \(\mathcal M_l^\alpha\)) to be the class of all associative rings \(R\) such that every homomorphic image of \(R\) contains a nonzero ideal (one-sided ideal, left ideal) in \(\mathcal M_\beta\) (\(\mathcal M^\beta\), \(\mathcal M_l^\beta\)) for some \(\beta<\alpha\). The class \(\{\mathcal M_\alpha\}\) (\(\{\mathcal M^\alpha\}\), \(\{\mathcal M_l^\alpha\}\)) forms a chain the union of which is equal to the lower radical class \(l(\mathcal M)\) (lower strong radical class \(ls(\mathcal M)\), lower left strong radical class \(ls_l(\mathcal M)\)) determined by \(\mathcal M\). The chain \(\{\mathcal M_\alpha\}\) is called the Kurosh's chain of \(\mathcal M\). The ADS-problem for the chains asks for classes whose Kurosh's chain stabilizes at any given step.NEWLINENEWLINE In this paper the authors give a historical outline of known results concerning the ADS-problem placing special emphasis on those that use accessible subrings, iterated maximal extensions of rings and completely normal rings. They list many old nontrivial questions related to the ADS-problem and put many new ones. They also give new examples of classes for which the Kurosh's chain stabilizes at any given step. In particular, they show that if \(P\) is a nonfilial completely normal ring such that if \(f\colon A\to B\) is a surjective homomorphism of nonzero accessible subrings of \(P\), then \(f\) is an isomorphism; \(\mathcal P\) is the class of all proper homomorphic images of all accessible subrings of \(P\) and \(\mathcal S=st(\mathcal P\cup\{R:R^2=0\})\), then, for every natural number \(n\), the set \(\mathcal P(n)\) of all precisely \(n\)-accessible subrings of \(P\) is nonempty. Moreover, for a nonempty set \(\mathcal A(n)\subseteq\mathcal P(n)\) and \(\mathcal N(n)=\mathcal S\cup\{\mathcal A(n)\}\) we have \(\mathcal N(n)_n=\mathcal N(n)^n\neq\mathcal N(n)_{n+1}=\mathcal N(n)^{n+1}=l(\mathcal N(n))=ls(\mathcal N(n))=st(\mathcal N(n))\). If \(\mathcal A(n)=\mathcal P(n)\), then the radical class \(l(\mathcal N(n))\) is hereditary.NEWLINENEWLINE Let \(R\) be a ring such that \(R=\mathrm{IME}(J)\) for every nonzero ideal \(J\) of \(R\). Moreover, let for all nonzero ideals \(B\) and \(C\) of \(R\), every surjective homomorphism \(f\colon B\to C\) be an isomorphism. Let \(\mathcal S\) be the lower radical determined by the class of all rings with zero multiplication and the class of all proper homomorphic images of all nonzero accessible subrings of \(R\). Let \(\mathcal A\) be a nonempty family of precisely \(n\)-accessible subrings of \(R\) for some natural number \(n\) and \(\mathcal M=\{\mathcal A\}\cup\mathcal S\). The authors show that then \(l(\mathcal M)=\mathcal M_{n+1}\) and \(R\in l(\mathcal M)\setminus\mathcal M_n\).NEWLINENEWLINE Let \(A\) be a semiprime ring such that \(pA=0\) for some prime number \(p\) and let \(A\) be a precisely \(n\)-accessible subring of a ring \(R\) for some positive integer \(n\) greater or equal to \(2\) and \(R=\mathrm{IME}(A)\). The author show that if \(A\) has no semiprime proper homomorphic image, or every proper semiprime homomorphic image of \(A\) is idempotent, then \(l(\mathcal M)=\mathcal M_{n+1}\), where \(\mathcal M\) is the class of all homomorphic images of \(A\).NEWLINENEWLINE As the grand finale, the authors prove that for every positive integer \(n\) there exists a ring \(A\) such that the class \(\mathcal M\) of all homomorphic images of \(A\) satisfies the condition \(l(\mathcal M)=\mathcal M_{n+1}\neq\mathcal M_n\). Namely, for \(n=1\), \(A\) is a nonzero idempotent ring; and for \(n\) greater or equal to \(2\), \(A=[x^2]+\mathbb Z_p[x]x^{2n}\), where \(p\) is a prime number and \(\mathbb Z_p\) denotes the field of \(p\)-elements.