Closure rings. (English) Zbl 1014.16019

Let \(R\) be a ring and let \(\circ\) denote the adjoint operation given by \(a\circ b=a+b-ab\) for all \(a,b\in R\). A ring \(R\) is said to be a closure ring if it has an additional unary operation \(C\) such that for all \(a,b\in R\): (1) \(C(0)=0\); (2) \(a\circ C(a)=C(a)\); (3) \(C(C(a))=C(a)\); (4) \(C(a)\circ C(b)=C(b)\circ C(a)\); (5) \(C(a)\circ C(b)=C(a\circ b)\circ C(b)\). If, moreover, \(a\circ b\circ C(a)=b\circ C(a)\), then \(R\) is said to be a strong closure ring. The first part presents some basic properties of such rings and the second one deals with the relations between normal filters and closed ideals. The third paragraph relates the closure operation on the ring \(R\) and on the matrix ring \(M_n(R)\) and on the polynomial ring \(R[S]\) (\(S\) is a set of indeterminates), respectively. In the last item the closure operations defined in terms of the lattice of central idempotents are investigated.


16N20 Jacobson radical, quasimultiplication
06A15 Galois correspondences, closure operators (in relation to ordered sets)
06E20 Ring-theoretic properties of Boolean algebras
16U99 Conditions on elements
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