0

I know that there is definitely module and vector space these two objects are similar. see the quotation below:

A module is abstractly very similar to a vector space, although in modules, coefficients are taken in rings that are much more general algebraic objects than the fields used in vector spaces. A module taking its coefficients in a ring is called a module over , or a R-module.

Question 1: I did not understand the meaning of line "general algebraic objects " in the above paragraph.

Question 2: What are other things that are similar or not similar between module and vectors space in terms of geometry?

  • In a general ring you don't necessarily have multiplicative inverses and may have zero divisors (in the integers taken modulo $6$ we have $2\times 3 \equiv 0$, for example). A vector space can be regarded as the special case of a module in which the scalars are taken from a field. – Mark Bennet Nov 02 '17 at 07:33
  • It's not always true that a module has a basis. – Ben P. Nov 02 '17 at 09:27
  • There are a lot of good answers at the duplicate that address a great deal of your questions. But as for Question 1 above, "general algebraic objects" just refers to "rings" which can behave a lot differently than fields in general. There is nothing more mysterious about the phrase. – rschwieb Nov 02 '17 at 13:26

0 Answers0