Some results on completability in commutative rings

Elementary matrix reduction over certain rings

“…By using Corollary 2.2 again, eRe is J-stable. ✷ Following Moore and Steger, a ring R is called a B-ring provided that a 1 R + · · · + a n R = R(n ≥ 3) with (a 1 , · · · , a n−2 ) J(R) =⇒ there exists b ∈ R such that a 1 R + · · · + a n−2 R + (a n−1 + a n b)R = R. Elementary properties of such rings have been studied in [9]. Surprisingly, we shall prove the classes of J-stable rings and B-rings coincide with each other.…”

A strong complement property of Dedekind domains

“…If 1 ^ /с g r, it follows that either a""i or a" does not belong to Ml^'^^. The Chinese Remainder Theorem guarantees beR such that b = = 0 {mod Ml^"-^) if а"_1фМ1''''\ b = ^modMf""^) if a"_^eMl^^\ for /c = = 1, 2,..., r. Hence a"_i + ba^ ф Ml'"'^^, к = 1,..., r, and if (a^,..., a"_2, «n-i + (For details, see [1], Lemma 3.3. )Since del,the induction hypothesis assures us of an (n -1) X (n -1) matrix D whose first row is a^, ..., a"_i + ba" and such that det D = d. и we let 71-2 [0 ... 0 1 then В is the desired matrix…”

A characterization of boolean spaces

“…S i s called a B-ring i f for each integer n 2 3 and each a , ..., 8 (. S such that (e , ..., e ) <^J(S) and 1 € (8., ..., 8 ) , there exists t € S such that 1 € (e , ..., 8 , 8 +t8 ) ; see [4] for details. Here, the notation (8., ..., a ) means the ideal of S generated by s , ..., 8…”

“…Let S be a commutative ring with identity. S is called an SB-ring if for each 8, a, d, e € S with 8 € (c, d, e) and a $ J{S) , i t follows that 8 € (a, d+te) for some t € S ; see [4] for details.…”

“…Hence, X i s a T-space and C(X) i s a Hermite ring. Since Hermite B-rings are SB-rings [4,Theorem 3 . 3 ] , i t follows that C(X) i s an SB-ring.…”