Shahid Rajaee Teacher Training University Journal of Discrete Mathematics and Its Applications 2981-0809 10 3 2025 09 01 Algebraically and geometrically closed of idempotents 283 291 2377 10.22061/jdma.2025.11872.1125 EN Ali Molkhasi Department of Mathematics, Faculty of Mathematical Sciences, Farhangian University of Tehran, Iran Journal Article 2025 03 30 Our aim in this article is to study algebraically and geometrically closed structures in a commutative ring with unity R. It is proved that the lattice of idempotents E of R is an algebraically closed lattice. We also show that if E is dense-in-itself, then E* is geometrically closed in Mod(T, A ). Finally, the relationship between an equicharacteristic regular local ring and an algebraically closed residue field is considered. algebraically closed geometrically closed idempotents https://jdma.sru.ac.ir/article_2377_42ce82a15a8171a521ba8a3f452cc73f.pdf