An Upper Bound for the w-Weak Global Dimension of Pullbacks

Let [Formula: see text] be a commutative ring with identity and [Formula: see text] an ideal of [Formula: see text]. We introduce and study the [Formula: see text]-weak global dimension [Formula: see text] of the factor ring [Formula: see text]. Let [Formula: see text] be a [Formula: see text]-linked extension of [Formula: see text], and we also introduce the [Formula: see text]-weak global dimension [Formula: see text] of [Formula: see text]. We show that the ring [Formula: see text] with [Formula: see text] is exactly a field and the ring [Formula: see text] with [Formula: see text] is exactly a [Formula: see text]. As an application, we give an upper bound for the [Formula: see text]-weak global dimension of a Cartesian square [Formula: see text]. More precisely, if [Formula: see text] is [Formula: see text]-linked over [Formula: see text], then [Formula: see text]. Furthermore, for a Milnor square [Formula: see text], we obtain [Formula: see text].

Arithmetical rings and Krull dimension

Let A be a commutative arithmetical ring. It is proved that the ring A has Krull dimension if and only if every factor ring of A is finite-dimensional and does not have idempotent proper essential ideals.

RADICAL RELATED TO SPECIAL ATOMS REVISITED

A semiprime ring $R$ is called a $\ast$-ring if the factor ring $R/I$ is in the prime radical for every nonzero ideal $I$ of $R$. A long-standing open question posed by Gardner asks whether the prime radical coincides with the upper radical $U(\ast _{k})$ generated by the essential cover of the class of all $\ast$-rings. This question is related to many other open questions in radical theory which makes studying properties of $U(\ast _{k})$ worthwhile. We show that $U(\ast _{k})$ is an N-radical and that it coincides with the prime radical if and only if it is complemented in the lattice $\mathbb{L}_{N}$ of all N-radicals. Along the way, we show how to establish left hereditariness and left strongness of important upper radicals and give a complete description of all the complemented elements in $\mathbb{L}_{N}$.

The Centralizer of an I-Matrix in M2(R/I), R a UFD

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 [Formula: see text] in M2(R/I) as the sum of two subrings 𝒮1 and 𝒮2 of M2(R/I), where 𝒮1 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 [Formula: see text], and the entries in 𝒮2 are intersections of certain annihilators of elements arising from the entries of [Formula: see text]. 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.