Sivák, Bohuslav The structure of the rings assigned to group varieties. (English) Zbl 0584.20017 Math. Slovaca 35, 169-173 (1985). The author investigates the following construction of R. S. Freese and R. McKenzie which assigns to each congruence-modular variety V some ring R(V). Let \(F_ 2\) be the V-free algebra generated by \(\{x,y\}\), \(\Gamma\) be the least congruence on \(F_ 2\) which identifies x and y, \(\pi\) : \(F_ 2\to F_ 2/[\Gamma,\Gamma]\) be the natural projection on the factor-algebra, \({\bar \Gamma}=\pi (\Gamma)\), \(R(V)=[y]{\bar \Gamma}\). The ring operations on R(V) are the following ones: \[ u(x,y)+v(x,y)=d(u(x,y),y,v(x,y)),\quad u(x,y)\cdot v(x,y)=u(v(x,y),y), \]\[ -u(x,y)=d(y,u(x,y),y),\quad 1=x,\quad 0=y. \] Here d is the ternary difference term in V. It is proved the following: 1) For varieties V of groups R(V) are homomorphic images of Z[p,q]; 2) If V is the variety of all groups then \(R(V)\cong Z[p,q]/(1-pq)\); 3) The assignment \(V\to R(V)\) is not injective. If K is the variety of groups determined by the identity \([[x,y],[z,t]]=1\), then \(R(V)\cong R(V\cap K)\) for each variety V of groups. Reviewer: S.R.Kogalovskij MSC: 20E10 Quasivarieties and varieties of groups 08B10 Congruence modularity, congruence distributivity 20F12 Commutator calculus Keywords:congruence-modular variety; free algebra; variety of groups × Cite Format Result Cite Review PDF Full Text: EuDML References: [1] FREESE R. S., McKENZIE R.: The commutator. an overview This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.