I have completed my coverage of ring homomorphisms: in this section, I enjoyed the theorem that stated that the intersection of all of the sub-fields of a field F is a sub-field of F that is isomorphic to either the ring Z_p (for some prime p) under addition and multiplication modulo p or the set of rational numbers under ordinary addition and multiplication (this depends on whether the multiplicative identity element of F, denoted as 1_F, has a finite order n [the smallest positive integer n that yields nx=0_F for all elements x in F is called the characteristic of F] or infinite order in the additive group (F,+)); I also enjoyed the ring isomorphism theorems and how the knowledge of homomorphisms/isomorphisms of groups could be utilized to help understand these theorems as they relate to rings; the embedding theorems were also very neat.
That said, I've completed my self-study of abstract algebra I and have definitely gained an appreciation for this material. I'm glad that I decided to explore this topic.
That said, I've completed my self-study of abstract algebra I and have definitely gained an appreciation for this material. I'm glad that I decided to explore this topic.
