This paper is primarily a survey of the results on essential covers of radical classes of rings. However, some problems are posed and some new results are presented.
Let \(R\) be a ring and \(K\) be an abstract class of rings. If \(X\vartriangleleft R\) and \(X\subseteq Y\), then \(X\) is \(R\)-essential in \(Y\) if \(0\leq B\vartriangleleft R\) and \(B\subseteq Y\) imply that \(B\cap X\neq 0\). If \(Y=R\) then \(X\) is said to be essential in \(R\) and \(R\) is said to be an essential extension of \(X\). The class consisting of all essential extensions of rings belonging to \(K\) is called the essential cover of \(K\).
If \(R\) is a ring, let \(P(R)\) denote the prime radical of \(R\). A subring \(A\) of \(R\) is said to be an \(n\)-accessible subring of \(R\) if there exists a chain \(A=B_1\vartriangleleft B_1\vartriangleleft\cdots\vartriangleleft B_m=R\) of subrings of \(R\).
The author uses the following notation:
\({\mathcal E}K\) – the essential cover of \(K\),
\(H^nK:=\{A\mid A\) is an \((n+1)\)-accessible subring of some ring in \(K\}\),
\(H^\infty K:=\bigcup^\infty_{n=1}H^nK\) (i.e., the hereditary closure of \(K\)),
\(UK:=\{R\mid R\) has no non-zero homomorphic image in \(K\}\),
\(U_rK\) – the upper radical determined by the class \(K\).
In this paper, the author considers the concept of “essentiality” in the context of radical theory. The paper is primarily expository in nature and emphasizes the historical development through a discussion of the connections and consequences of various known results.
The author also presents the following new result: Theorem. Let \(\theta\) be a radical such that \(P\subseteq\theta\). Then: (i) \(H{\mathcal E}\theta\subseteq{\mathcal E}H\theta\) and (ii) \(H{\mathcal E}\theta\) and \({\mathcal E}H\theta\) are regular classes.
As a result of this theorem, we have: Corollary. Let \(\theta\) be a radical such that \(P\subseteq\theta\). We have the following chains of radicals: (i) \(U{\mathcal E}H^\infty\theta\subseteq U{\mathcal E}H\theta\subseteq UH{\mathcal E}\theta\), (ii) \(U_r{\mathcal E}\theta\subseteq UH^\infty{\mathcal E}\theta\subseteq UH{\mathcal E}\theta\), (iii) If \(\theta\) is supernilpotent, then \(U{\mathcal E}H^\infty\theta=U_r{\mathcal E}\theta=UH{\mathcal E}\theta=U{\mathcal E}\theta\).
The paper closes with a final problem: Find necessary and/or sufficient conditions on a radical \(\theta\) such that: (i) \(U{\mathcal E}H\theta\subseteq U_r{\mathcal E}\theta\subseteq UH{\mathcal E}\theta\) or (ii) \(U{\mathcal E}H\theta=U_r{\mathcal E}\theta\) or (iii) \(U_r{\mathcal E}\theta=UH{\mathcal E}\theta\).