It's all a generalization of Linear Algebra!

Linear Algebra: The study of vector spaces over fields.

Examples	0	R	RxR	RxRxR
Pictures
Generators	0	1	[1] [0] [0], [1]	[1] [0] [0] [0], [1], [0] [0] [0] [1]

Commutative Algebra: The study of modules over commutative rings.

Examples:

• Vector spaces are modules over fields (the field doesn't have to be R, it could be Q, C, Z_p, etc.)
  
  
• Abelian groups are modules over the ring of integers, Z
  
  
• There are many other rings and modules.

We know how to decompose a vector space into products and how to give generators for a vector space. What about modules?