Example on Z-modules .... Dummit & Foote, Page 339 ....

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In summary, The module axioms demonstrate that the only possible action of $\mathbb{Z}$ on $A$ making it a unital $\mathbb{Z}$-module is by multiplying each element in $A$ by a natural number $n$, which is achieved by adding $a$ to itself $n$ times. This is proven by using the module axioms to show that this is the only possible value for $na$, taking into account the cases where $n=0$ and $n<0$.
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I am reading Dummit and Foote's book: "Abstract Algebra" (Third Edition) ...

I am currently studying Chapter 10: Introduction to Module Theory ... ...

I need some help with an aspect of Dummit and Foote's example on Z-modules in Section 10.1 Basic Definitions and Examples ... ... Dummit and Foote's example on Z-modules reads as follows:
https://www.physicsforums.com/attachments/8001In the above example we read the following:

" ... ... This definition of an action on the integers on \(\displaystyle A\) makes \(\displaystyle A\) into a \(\displaystyle \mathbb{Z}\)-module, and the module axioms show that this is the only possible action of \(\displaystyle \mathbb{Z}\) on \(\displaystyle A\) making it a (unital) \(\displaystyle \mathbb{Z}\)-module ... ... "Can someone please explain how/why the module axioms demonstrate that this is the only possible action of \(\displaystyle \mathbb{Z}\) on \(\displaystyle A\) making it a (unital) \(\displaystyle \mathbb{Z}\)-module ... ... ? How do we know it is the only possible such action ... ... ?Hope someone can help ...

Peter
 
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If $n>0$ then $n = 1+1+\ldots + 1$ (property of $\Bbb{Z}$). Therefore $$na = (1+1+\ldots + 1)a = 1a + 1a + \ldots + 1a = a+a+\ldots + a,$$ using the module axioms. So that is the only possible value for $na$.The cases $n=0$ and $n<0$ work in a similar way, starting from the facts that $0 = 1 + (-1)$ and $-n = (-1)n$.
 

FAQ: Example on Z-modules .... Dummit & Foote, Page 339 ....

1. What are Z-modules?

Z-modules are modules over the ring of integers, Z. They are also known as abelian groups, and they are an important algebraic structure that is used in abstract algebra and number theory.

2. How are Z-modules different from other modules?

Z-modules have the additional property that the elements in the module commute with each other under multiplication. This means that the order of multiplication does not matter, unlike in other modules where it may affect the result.

3. What is the significance of Z-modules in algebraic structures?

Z-modules play a crucial role in the study of abstract algebra and number theory. They are used to understand the properties of integers and their relationships with other mathematical structures. Z-modules are also important in the study of vector spaces and their subspaces.

4. Can you provide an example of a Z-module?

An example of a Z-module is the set of all integers, Z, with addition as the operation and multiplication defined as scalar multiplication. In this case, the elements of Z commute with each other under multiplication, making it a Z-module.

5. How are Z-modules related to group theory?

Z-modules are closely related to abelian groups, which are also known as commutative groups. Group theory is the study of abstract algebraic structures, and Z-modules, being commutative groups, are an important topic in this field. Furthermore, Z-modules can be used to study the properties of other groups, such as finite cyclic groups.

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