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We show that any quasi-Gorenstein deformation of a 3-dimensional quasi-Gorenstein Buchsbaum local ring with I -invariant 1 admits a maximal Cohen-Macaulay module, provided it is a quotient of a Gorenstein ring. Such a class of rings includes two instances of unique factorization domains constructed by Marcel-Schenzel and by Imtiaz-Schenzel, respectively. Apart from this result, motivated by the small Cohen-Macaulay conjecture in prime characteristic, we examine a question about when the Frobenius pushforward F ⁎ e (M) of an R -module M comprises a maximal Cohen-Macaulay direct summand in both local and graded cases. [ABSTRACT FROM AUTHOR]

We reduce Hochster's Canonical Element Conjecture (theorem since 2016) to a localization problem in a characteristic free way. We prove the validity of a new variant of the Canonical Element Theorem (CET) and explain how a characteristic free deduction of the new variant from the original CET would provide us with a characteristic free proof of the CET. We also show that the Balanced Big Cohen-Macaulay Module Theorem can be settled by a characteristic free proof if the big Cohen-Macaulayness of Hochster's modification module can be deduced from the existence of a maximal Cohen-Macaulay complex in a characteristic free way. [ABSTRACT FROM AUTHOR]

We give several criteria for a ring to be a UFD, including generalizations of some criteria due to P. Samuel. These criteria are applied to construct, for any field k , (1) a Z -graded non-noetherian rational UFD of dimension 3 over k , and (2) k -affine rational UFDs defined by trinomial relations. [ABSTRACT FROM AUTHOR]

Additive decompositions over finite fields were extensively studied by Brawely and Carlitz. In this paper, we study the additive decomposition of polynomials over unique factorization domains. [ABSTRACT FROM AUTHOR]

Let R be a commutative unital ring, and let Spec (R) : = P (R) be the collection of prime ideals of R. Further, let N P (R) denote the collection of proper nonprime ideals of R and set N P (R) 1 : = N P (R) ∪ { R }. In this note, we investigate some closure properties relative to these collections of ideals with respect to ideal addition and multiplication. Among the main objects of our study are rings R for which P (R) + N P (R) ⊆ P (R) , N P (R) 1 + N P (R) 1 ⊆ N P (R) 1 , or P (R) · P (R) ⊆ P (R). Directions for further research are indicated. Communicated by Sudarshan Kumar Sehgal [ABSTRACT FROM AUTHOR]

We propose a divisibility criterion for elements of a generic Unique Factorization Domain. As a consequence, we obtain a general divisibility criterion for polynomials over Unique Factorization Domains. The arguments can be used in basic algebra courses and are suitable for building classroom/homework activities for college and high school students. [ABSTRACT FROM AUTHOR]

Let d be a square-free positive integer, σ = 1 if d ≡ 1 (mod 4) and σ = 2 otherwise. Let p , b , c and x 0 be integers, where p is a prime. Suppose that b 2 − 4 p c = t 2 d for some integer t ≥ 1. Suppose that there exist integers x and y such that p = | x 2 − d y 2 |. We prove that if | p n 2 + b n + c | is 1 or a prime for all integers n with x 0 ≤ n ≤ x 0 + σ d 3 , then the class number of the field ℚ (d) is 1. [ABSTRACT FROM AUTHOR]

In this paper we show that if I is an ideal of a commutative semigroup C such that the separator SepI of I is not empty then the factor semigroup $$S=C/P_I$$ ( $$P_I$$ is the main congruence of C defined by I) satisfies Condition $$(*)$$ : S is a commutative monoid with a zero; The annihilator A( s) of every non identity element s of S contains a non zero element of S; $$A(s)=A(t)$$ implies $$s=t$$ for every $$s, t\in S$$ . Conversely, if $$\alpha $$ is a congruence on a commutative semigroup C such that the factor semigroup $$S=C/\alpha $$ satisfies Condition $$(*)$$ then there is an ideal I of C such that $$\alpha =P_I$$ . Using this result for the multiplicative semigroup $$D_{mult}$$ of a unique factorization domain D, we show that $$P_{J(m)}=\tau _m$$ for every nonzero element $$m\in D$$ , where J( m) denotes the ideal of D generated by m, and $$\tau _m$$ is the relation on D defined by $$(a, b)\in \tau _m$$ if and only if $$gcd(a, m)\sim gcd(b, m)$$ ( $$\sim $$ is the associate congruence on $$D_{mult}$$ ). We also show that if a is a nonzero element of a unique factorization domain D then $$d(a)=|D'/P_{J([a])}|$$ , where d( a) denotes the number of all non associated divisors of a, $$D'=D/\sim $$ , and [ a] denotes the $$\sim $$ -class of $$D_{mult}$$ containing a. As an other application, we show that if d is one of the integers $$-1$$ , $$-2$$ , $$-3$$ , $$-7$$ , $$-11$$ , $$-19$$ , $$-43$$ , $$-67$$ , $$-163$$ then, for every nonzero ideal I of the ring R of all algebraic integers of an imaginary quadratic number field $${\mathbb Q}[\sqrt{d}]$$ , there is a nonzero element m of R such that $$P_I=\tau _m$$ . [ABSTRACT FROM AUTHOR]

We investigate the problem on the validity of the Lazard theorem analogue (or the Kronecker–Vahlen theorem), which states that the least remainder Euclidean Algorithm (EA) has the shortest length over any other versions of EA, in unique factorization domains. There is obtained the existence criterion to represent a fixed element of the fractions field of a unique factorization domain in the form of a continued fraction of a fixed length. This criterion enables to obtain a formula for the length of the least remainder (on norm) EA as a function of elements, with respect to which EA is applied. This result gives us the class T of unique factorization domains, for which the Lazard theorem analogue is valid. We propose algorithms to check whether the given unique factorization domain belongs to the class T . We find the necessary and sufficient conditions under which the number of steps in the worst case of the least remainder EA grows not faster than logarithm. In particular, these results hold for the least remainder EA in any Euclidean quadratic domain. We provide counterexamples, which show the essentiality of the conditions in the obtained theorems. [ABSTRACT FROM AUTHOR]

In this paper, we give an elementary proof of the fact that the rings Z h 1+√ −d 2 i are unique factorization domains for the values d = 3, 7, 11, 19, 43, 67, 163. While the result in itself is well known, our proof is new and completely elementary and uses neither the Minkowski convex body theorem, nor the Dedekind and Hasse theorems. Furthermore, it does not use either the theory of algebraic integers, or the theory of Noetherian rings. It only uses basic notions from the theory of commutative rings. [ABSTRACT FROM AUTHOR]

Let R be a locally quasi-unmixed domain, a , b 1 , … , b n an asymptotic sequence in R , I = ( a , b 1 , … , b n ) R and S = R [ b 1 / a , … , b n / a ] = R [ I / a ] , the monoidal transform of R with respect to I . It is shown that S is a locally quasi-unmixed domain, a , b 1 / a , … , b n / a is an asymptotic sequence in S and there is a one-to-one correspondence between the asymptotic primes A ˆ ⁎ ( I ) of I and the asymptotic primes A ˆ ⁎ ( a S ) of a S = I S . Moreover, if a , b 1 , … , b n is an R -sequence, this extends to a one-to-one correspondence between Ass R ( R / I ) and Ass S ( S / a S ) . In the case that R is a unique factorization domain, the height one prime ideals of S are examined to determine how far S is from being a UFD. A complete description is given of which height one prime ideals P of S are principal or have a principal primary ideal in the case that P ∩ R has height 1. If the prime divisors of a satisfy a mild condition, a similar description is given in the case that P ∩ R has height >1. These are applied to give similar results for the Rees ring R [ 1 / t , I t ] where t is an indeterminate. [ABSTRACT FROM AUTHOR]

The concept of an I-matrix in the full 2 × 2 matrix ring M2(R/I), where R is an arbitrary UFD and I is a nonzero ideal in R, is introduced. We obtain a concrete description of the centralizer of an I-matrix in M2(R/I) as the sum of two subrings 풮1 and 풮2 of M2(R/I), where 풮1 is the image (under the natural epimorphism from M2(R) to M2(R/I)) of the centralizer in M2(R) of a pre-image of , and the entries in 풮2 are intersections of certain annihilators of elements arising from the entries of . It turns out that if R is a PID, then every matrix in M2(R/I) is an I-matrix. However, this is not the case if R is a UFD in general. Moreover, for every factor ring R/I with zero divisors and every n ≥ 3, there is a matrix for which the mentioned concrete description is not valid. [ABSTRACT FROM AUTHOR]

The concept of anI-matrix in the full 2 × 2 matrix ringM2(R/I), whereRis an arbitrary UFD andIis a nonzero ideal inR, was introduced in a Marais article that is to be published [10]. Moreover, a concrete description of the centralizer of anI-matrixinM2(R/I) as the sum of two subrings 𝒮1and 𝒮2ofM2(R/I) was also given, where 𝒮1is the image (under the natural epimorphism fromM2(R) toM2(R/I)) of the centralizer inM2(R) of a pre-image of, and where the entries in 𝒮2are intersections of certain annihilators of elements arising from the entries of. In the present paper, we obtain results for the case whenIis a principal ideal⟨k⟩,k ∈ Ra nonzero nonunit. Mainly we solve two problems. Firstly we find necessary and sufficient conditions for when 𝒮1 ⊆ 𝒮2, for when 𝒮2 ⊆ 𝒮1and for when 𝒮1 = 𝒮2. Secondly we provide a formula for the number of elements in the centralizer offor the case whenR/⟨k⟩is finite. [ABSTRACT FROM AUTHOR]

We prove the following fact: If finitely many elements p, p,..., p of a unique factorization domain are given such that the greatest common divisor of each pair ( p, p) can be expressed as a linear combination of p and p, then the greatest common divisor of all the p's can also be expressed as a linear combination of p,..., p. We prove an analogous statement in general commutative rings. [ABSTRACT FROM AUTHOR]

In this article, we develop equivalent conditions for a certain class of monoidal transform to inherit either the property of being a completely integrally closed domain that satisfies the ascending chain condition on principal ideals, the property of being a Mori domain, the property of being a Krull domain, or the property of being a unique factorization domain, respectively. Such a class of monoidal transform is given in terms of an (analytically) independent set that forms a prime ideal in the base domain. Characterizations are provided illustrating the necessity of the “prime ideal” hypothesis when the base domain is a Noetherian unique factorization domain. [ABSTRACT FROM AUTHOR]

In this article, we raise the question as to what conditions permit a simple overring of a domain R —that is, a domain of the form for some f, g ∈ R such that g ≠ 0—to inherit the ascending chain condition on principal ideals from R . Our main theorem reveals that, if g is a prime element of R , the complete answer can be found by considering the Archimedean property. We then, in turn, use this theorem to establish equivalent conditions for a certain class of simple overring to inherit the property of being a unique factorization domain from R . [ABSTRACT FROM AUTHOR]

Let (T, M) be a complete local normal integral domain containing the rationals such that | T / M | = c where c is the cardinality of the real numbers. Let p be a non-maximal prime ideal of T such that T p is a regular local ring. We construct a local Unique Factorization Domain (UFD) A such that the M -adic completion of A is T , p is maximal in the generic formal fiber and all fibers of A are geometrically regular except for those over some height one prime ideals. [ABSTRACT FROM AUTHOR]

In studying unique factorization of domains we encountered a property of ideals. Using that we define the notion of almost prime ideals and prove that in Noetherian domains almost prime ideals are primary. We also prove that in a regular domain almost primes are precisely primes. Further, we define strictly nonprime ideals and study some inter relations between almost prime ideals, strictly nonprime ideals and factorization of ideals. [ABSTRACT FROM AUTHOR]