Combining Euclidean and adequate rings

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

Không thể tạo bản xem trước, hãy bấm tải xuống
TÀI LIỆU MỚI ĐĂNG
Đã phát hiện trình chặn quảng cáo AdBlock
Trang web này phụ thuộc vào doanh thu từ số lần hiển thị quảng cáo để tồn tại. Vui lòng tắt trình chặn quảng cáo của bạn hoặc tạm dừng tính năng chặn quảng cáo cho trang web này.