From Real Vector Spaces to Modules in 7 Lessons

Perhaps millions of undergraduate students (in Mathematics, Physics, Chemistry, Medicine, Education, Biology, Sociology and maybe many others) are taught a very popular Math course, usually called Linear Algebra. In doing so, they first meet the simplest and most friendly algebra structure: the real vector space. Only a few of them choose to continue (mostly students with a Science major) and are taught something called Advanced Linear Algebra, which essentially changes the set of scalars (initially real numbers) by an arbitrary division ring. Finally, better can be done: scalars can be taken in an arbitrary ring R with identity. This way R-modules over rings are defined and this is customarily a graduate course for Math students. This booklet is recommended for undergraduate (and graduate) students, after completing a Linear Algebra Course.