where θ = 1 + √ d 2 if d mod 4 = 1 and θ = √ d if d mod 4 = 2, 3. The purpose of this paper is to compute the extended gcd of to quadratic integers in ring Z[θ]. We assume throughout that Z[θ] is principal ideal ring, but not necessarily an euclidean ring. If [a, b+ cθ] is the module {ax+(b+ cθ)y, x, y ∈ Z}, it can be shown [3] that I is an ideal of Z[θ] if and only if I = [a, b+ cθ]; where a, ...

The knowledge about education to build an ideal Christian family obtained by Generation Z is not good enough, so the author gives purpose of writing discuss crisis in household, rebuild household and provide values generation preventing divorce. uses descriptive literature research method data provides results: How a harmonious can be dynamically interwoven if we base our development on strong ...

Let $R$ be an associative ring with unity. An element $x \in R$ is called $\mathbb{Z}G$-clean if $x=e+r$, where $e$ is an idempotent and $r$ is a $\mathbb{Z}G$-regular element in $R$. A ring $R$ is called $\mathbb{Z}G$-clean if every element of $R$ is $\mathbb{Z}G$-clean. In this paper, we show that in an abelian $\mathbb{Z}G$-regular ring $R$, the $Nil(R)$ is a two-sided ideal of $R$ and $\fra...

Recently many researchers have worked on problems connected with various graded algebras associated to an ideal I in a local ring R. Two algebras in particular have received the most attention: the associated graded algebra of Z, gr,( R) = R/Z@ Z/Z’@ ... , and the Rees algebra of Z, defined to be R[Zt]. Brodmann [2] and Goto and Shimoda [9] have studied the local cohomology of R[Zr] for certain...

In this paper we show that if K is an incline with multiplicative identity and I is an r-ideal of K containing a unit u, then I = K. Moreover, we show that in a non-zero incline K with multiplicative identity and zero element, every proper r-ideal in K is contained in a maximal r-ideal of K. Z. Q. Cao, K. H. Kim and F. W. Roush [3] introduced the notion of incline algebras in their book, Inclin...

BACKGROUND The goal of this study was to identify the effect of the change in body mass index (BMI) from childhood to adulthood on body image satisfaction at 23 years of age in members of the 1982 Pelotas Birth Cohort in Pelotas, RS, Brazil. METHODS The study used data from the 1986 and 2004-5 follow-up studies. Body shape satisfaction was evaluated using the Stunkard scale. Body shape dissat...

In this paper, we introduce a method by which we can find a close connection between the set of prime $z$-ideals of $C(X)$ and the same of $C(Y)$, for some special subset $Y$ of $X$. For instance, if $Y=Coz(f)$ for some $fin C(X)$, then there exists a one-to-one correspondence between the set of prime $z$-ideals of $C(Y)$ and the set of prime $z$-ideals of $C(X)$ not containing $f$. Moreover, c...

in this article‎, ‎we have characterized ideals in $c(x)$ in which‎ ‎every ideal is also an ideal (a $z$-ideal) of $c(x)$‎. ‎motivated by‎ ‎this characterization‎, ‎we observe that $c_infty(x)$ is a regular‎ ‎ring if and only if every open locally compact $sigma$-compact‎ ‎subset of $x$ is finite‎. ‎concerning prime ideals‎, ‎it is shown that‎ ‎the sum of every two prime (semiprime) ideals of e...

Remark (Ideals versus subrings): It is worthwhile to compare these two notions; they are related, but with subtle and important differences. Both an ideal I and a subring S of a ring R are subsets of R which are subgroups under addition and are stable under multiplication. However, each has an additional property: for an ideal it is the absorption property (IR2). For instance, the integers Z ar...

Let A be the ring obtained by localizing the polynomial ring κ[X, Y, Z, W ] over a field κ at the maximal ideal (X, Y, Z, W) and modulo the ideal (XW − Y Z). Let p be the ideal of A generated by X and Y. We study the module structure of a minimal injective resolution of A/p in details using local cohomology. Applications include the description of Ext i A (M, A/p), where M is a module construct...