Categoria: didattica algebra 4 2015-2016

In the second lecture we recall basic facts from module theory: definition of module, and module homomorphisms, submodule, I II and III homomorphism theorems, direct sum and product of modules, free modules and basis.

Reference: Rotman “Advanced modern algebra” 7.1 and 7.4

In our first lecture we recall previous notions about rings and ideals: definitions, irreducible and prime and nilpotent elements (in particular in polynomial rings); ideals, prime and maximal ideals; quotient rings and bijection between ideals; I II and III omomorphism theorem for ideals; eucliden rings, PID, UFD; ideals of and , the nipotent elements are the intersection of all prime ideals, definition of the Jacobson radical (in A[x] the Jacobson radical is the same as the nilpotents), definition of the radical of an ideal, prove that the radical of I is the intersection of all primes containing I, idempotent elements, Pierce decomposition, Boolean rings (Boolean domain is Z_2, every prime ideal is maximal, etc).