We shall prove that an E-adequate ring is an elementary divisor ring if and only if it is a Hermite ring. Elementary matrix reduction over such rings is also studied. We thereby generalize Domsha, Vasiunyk, and Zabavsky’stheorems to a much wider class of rings. | Turk J Math (2016) 40: 506 – 516 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Combining Euclidean and adequate rings Huanyin CHEN1,∗, Marjan SHEIBANI2 Department of Mathematics, Hangzhou Normal University, Hangzhou, China 2 Faculty of Mathematics, Statistics and Computer Science, Semnan University, Semnan, Iran 1 Received: • Accepted/Published Online: • Final Version: Abstract: We combine Euclidean and adequate rings, and introduce a new type of ring. A ring R is called an E-adequate ring provided that for any a, b ∈ R such that aR + bR = R and c ̸= 0 there exists y ∈ R such that (a + by, c) is an E-adequate pair. We shall prove that an E-adequate ring is an elementary divisor ring if and only if it is a Hermite ring. Elementary matrix reduction over such rings is also studied. We thereby generalize Domsha, Vasiunyk, and Zabavsky’s theorems to a much wider class of rings. Key words: Euclidean rings, adequate rings, elementary divisor rings, elementary matrix reduction 1. Introduction Throughout this paper, all rings are commutative with an identity. A matrix A (not necessarily square) over a ring R admits diagonal reduction if there exist invertible matrices P and Q such that P AQ is a diagonal matrix (dij ), for which dii is a divisor of d(i+1)(i+1) for each i . A ring R is an elementary divisor ring provided that every matrix over R admits a diagonal reduction. A ring R is a Hermite ring if every 1 × 2 matrix over R admits a diagonal reduction. As is well known, a ring R is Hermite if and only if for all a, b ∈ R there exist a1 , , b1 ∈ R such that a = a1 d, b = b1 d and a1 R + b1 R = R ([10, Theorem ]). After Kaplansky’s work on elementary divisor rings without zero divisors, Gillman and Henriksen proved that Theorem [10, Theorem ] A ring R is an elementary divisor ring if and only if (1) R is a Hermite ring; (2) For all a1
