What Are the Key Properties and Questions About Free Modules?

[gianeshwar]

We define basis as:
Let M be a module over a ring R with unity and let S be a subset of M. Then S is called a basis of M if
1. M is generated by S 2. S is linearly independent set.

Also we define free module as
An R module M is called a free module if there exists a subset S of M s.t.S generates M and S is linearly independent set.
NOW my QUESTIONS are :
1.Does a free module require unity?
2.Can a free module be basis?
3.Please tell me also the difference between the notations {0},{},(0) in reference to modules.

* I am still a learner in this area of mathematics.

 

[micromass]

We define basis as:
Let M be a module over a ring R with unity and let S be a subset of M. Then S is called a basis of M if
1. M is generated by S 2. S is linearly independent set.

Also we define free module as
An R module M is called a free module if there exists a subset S of M s.t.S generates M and S is linearly independent set.
NOW my QUESTIONS are :
1.Does a free module require unity?

You mean whether the underlying ring $R$ needs to have a unit? No, that's not necessary to define free module.

2.Can a free module be basis?

No, a free module always has a basis, but it never is a basis. A free module is the same as a vector space over a field. A vector space always has a basis, but I'm sure you know a vector space never is a basis.

3.Please tell me also the difference between the notations {0},{},(0) in reference to modules.

The set $\{0\}$ is the set with as only element $0$.
The set $\{\}$ is the empty set $\emptyset$.
The set $(0)$ is the ideal generated by the element $0$. It coincides with $\{0\}$.

 

[gufiguer]

Search about "free objects" in a category (in category theory).