Further Remarks on the Operational Nonterminal Complexity

For a regular language [Formula: see text], let [Formula: see text] be the minimal number of nonterminals necessary to generate [Formula: see text] by right linear grammars. Moreover, for natural numbers [Formula: see text] and an [Formula: see text]-ary regularity preserving operation [Formula: see text], let the range [Formula: see text] be the set of all numbers [Formula: see text] such that there are regular languages [Formula: see text] with [Formula: see text] for [Formula: see text] and [Formula: see text]. We show that, for the circular shift operation [Formula: see text], [Formula: see text] is infinite for all [Formula: see text], and we completely determine the set [Formula: see text]. Moreover, we give a precise range for the left right quotient and a partial result for the left quotient. Furthermore, we add some values to the range for the operation intersection which improves the result of [2].

The classical left regular left quotient ring of a ring and its semisimplicity criteria

Let [Formula: see text] be a ring, [Formula: see text] and [Formula: see text] be the set of regular and left regular elements of [Formula: see text] ([Formula: see text]). Goldie’s Theorem is a semisimplicity criterion for the classical left quotient ring [Formula: see text]. Semisimplicity criteria are given for the classical left regular left quotient ring [Formula: see text]. As a corollary, two new semisimplicity criteria for [Formula: see text] are obtained (in the spirit of Goldie).

New criteria for a ring to have a semisimple left quotient ring

Goldie's Theorem (1960), which is one of the most important results in Ring Theory, is a criterion for a ring to have a semisimple left quotient ring. The aim of the paper is to give four new criteria (using a completely different approach and new ideas). The first one is based on the recent fact that for an arbitrary ring R the set ℳ of maximal left denominator sets of R is a non-empty set [V. V. Bavula, The largest left quotient ring of a ring, preprint (2011), arXiv:math.RA:1101.5107]: Theorem (The First Criterion). A ring R has a semisimple left quotient ring Q iff ℳ is a finite set, ⋂S∈ℳ ass (S) = 0 and, for each S ∈ ℳ, the ring S-1R is a simple left Artinian ring. In this case, Q ≃ ∏S∈ℳ S-1R. The Second Criterion is given via the minimal primes of R and goes further than the First one in the sense that it describes explicitly the maximal left denominator sets S via the minimal primes of R. The Third Criterion is close to Goldie's Criterion but it is easier to check in applications (basically, it reduces Goldie's Theorem to the prime case). The Fourth Criterion is given via certain left denominator sets.

Characterizations of left orders in left Artinian rings

Small (1966), Robson (1967), Tachikawa (1971) and Hajarnavis (1972) have given different criteria for a ring to have a left Artinian left quotient ring. In this paper, three more new criteria are given.