An alternative perspective on projectivity of modules

Similar to the idea of relative projectivity, we introduce the notion of relative subprojectivity, which is an alternative way to measure the projectivity of a module. Given modules $M$ and $N$, $M$ is said to be {\em $N$-subprojective} if for every epimorphism $g:B \rightarrow N$ and homomorphism $f:M \rightarrow N$, then there exists a homomorphism $h:M \rightarrow B$ such that $gh=f$. For a module $M$, the {\em subprojectivity domain of $M$} is defined to be the collection of all modules $N$ such that $M$ is $N$-subprojective. A module is projective if and only if its subprojectivity domain consists of all modules. Opposite to this idea, a module $M$ is said to be {\em subprojectively poor}, or {\em $sp$-poor} if its subprojectivity domain is as small as conceivably possible, that is, consisting of exactly the projective modules. Properties of subprojectivity domains and $sp$-poor modules are studied. In particular, the existence of an $sp$-poor module is attained for artinian serial rings.Comment: Dedicated to the memory of Francisco Raggi; v2 some editorial changes. 'Right hereditary right perfect' replaced by the (equivalent) condition 'right hereditary semiprimary'; v3 a mistake corrected in the statements of Propositions 3.8 and 3.

Higher education students’ perceptions of ChatGPT: A global study of early reactions

© 2025 Ravšelj et al.The paper presents the most comprehensive and large-scale global study to date on how higher education students perceived the use of ChatGPT in early 2024. With a sample of 23,218 students from 109 countries and territories, the study reveals that students primarily used ChatGPT for brainstorming, summarizing texts, and finding research articles, with a few using it for professional and creative writing. They found it useful for simplifying complex information and summarizing content, but less reliable for providing information and supporting classroom learning, though some considered its information clearer than that from peers and teachers. Moreover, students agreed on the need for AI regulations at all levels due to concerns about ChatGPT promoting cheating, plagiarism, and social isolation. However, they believed ChatGPT could potentially enhance their access to knowledge and improve their learning experience, study efficiency, and chances of achieving good grades. While ChatGPT was perceived as effective in potentially improving AI literacy, digital communication, and content creation skills, it was less useful for interpersonal communication, decision-making, numeracy, native language proficiency, and the development of critical thinking skills. Students also felt that ChatGPT would boost demand for AI-related skills and facilitate remote work without significantly impacting unemployment. Emotionally, students mostly felt positive using ChatGPT, with curiosity and calmness being the most common emotions. Further examinations reveal variations in students’ perceptions across different socio-demographic and geographic factors, with key factors influencing students’ use of ChatGPT also being identified. Higher education institutions’ managers and teachers may benefit from these findings while formulating the curricula and instructions/regulations for ChatGPT use, as well as when designing the teaching methods and assessment tools. Moreover, policymakers may also consider the findings when formulating strategies for secondary and higher education system development, especially in light of changing labor market needs and related digital skills development.The authors acknowledge the financial support from the Slovenian Research and Innovation Agency (research core funding No. P5-0093 and project No. Z5-4569)

Higher education students' perceptions of ChatGPT: A global study of early reactions

The paper presents the most comprehensive and large-scale global study to date on how higher education students perceived the use of ChatGPT in early 2024. With a sample of 23,218 students from 109 countries and territories, the study reveals that students primarily used ChatGPT for brainstorming, summarizing texts, and finding research articles, with a few using it for professional and creative writing. They found it useful for simplifying complex information and summarizing content, but less reliable for providing information and supporting classroom learning, though some considered its information clearer than that from peers and teachers. Moreover, students agreed on the need for AI regulations at all levels due to concerns about ChatGPT promoting cheating, plagiarism, and social isolation. However, they believed ChatGPT could potentially enhance their access to knowledge and improve their learning experience, study efficiency, and chances of achieving good grades. While ChatGPT was perceived as effective in potentially improving AI literacy, digital communication, and content creation skills, it was less useful for interpersonal communication, decision-making, numeracy, native language proficiency, and the development of critical thinking skills. Students also felt that ChatGPT would boost demand for AI-related skills and facilitate remote work without significantly impacting unemployment. Emotionally, students mostly felt positive using ChatGPT, with curiosity and calmness being the most common emotions. Further examinations reveal variations in students' perceptions across different socio-demographic and geographic factors, with key factors influencing students' use of ChatGPT also being identified. Higher education institutions' managers and teachers may benefit from these findings while formulating the curricula and instructions/regulations for ChatGPT use, as well as when designing the teaching methods and assessment tools. Moreover, policymakers may also consider the findings when formulating strategies for secondary and higher education system development, especially in light of changing labor market needs and related digital skills development