Search

In this paper, our objects of interest are Hopf Galois extensions (e.g., Hopf algebras, Galois field extensions, strongly graded algebras, crossed products, principal bundles, etc.) and families of noncommutative rings (e.g., skew polynomial rings, PBW extensions and skew PBW extensions, etc.) We collect and systematize questions, problems, properties and recent advances in both theories by explicitly developing examples and doing calculations that are usually omitted in the literature. In particular, for Hopf Galois extensions we consider approaches from the point of view of quantum torsors (also known as quantum heaps) and Hopf Galois systems, while for some families of noncommutative rings we present advances in the characterization of ring-theoretic and homological properties. Every developed topic is exemplified with abundant references to classic and current works, so this paper serves as a survey for those interested in either of the two theories. Throughout, interactions between both are presented.

Gallbladder mucocele is characterized by hyperplasia of the gallbladder epithelium, increased mucus production, accumulation, and densification of the bile content, which can lead to biliary obstruction, necrosis, and rupture of the gallbladder wall. Its finding may be accidental or related to symptoms. A retrospective study (2016-2019) was carried out based on abdominal ultrasound examinations in dogs, correlating aspects of the gallbladder and biliary system in the mucocele with existing comorbidities. Thirty dogs diagnosed with biliary mucocele were evaluated, of which 46.66% had the disease at an early stage, and 53.33% showed a more advanced stage. Of these, 66.66% were related to endocrinopathies and hyperadrenocorticism. Signs of extrahepatic bile duct obstruction and biliary peritonitis were observed in two animals. Due to their potential risk of complications, follow-up ultrasound assessments are indicated in cases that opt for clinical treatment, not excluding the need for surgical intervention.

Gallbladder mucocele is characterized by hyperplasia of the gallbladder epithelium, increased mucus production, accumulation, and densification of the bile content, which can lead to biliary obstruction, necrosis, and rupture of the gallbladder wall. Its finding may be accidental or related to symptoms. A retrospective study (2016-2019) was carried out based on abdominal ultrasound examinations in dogs, correlating aspects of the gallbladder and biliary system in the mucocele with existing comorbidities. Thirty dogs diagnosed with biliary mucocele were evaluated, of which 46.66% had the disease at an early stage, and 53.33% showed a more advanced stage. Of these, 66.66% were related to endocrinopathies and hyperadrenocorticism. Signs of extrahepatic bile duct obstruction and biliary peritonitis were observed in two animals. Due to their potential risk of complications, follow-up ultrasound assessments are indicated in cases that opt for clinical treatment, not excluding the need for surgical intervention.