How Is the Primary Decomposition Theorem Known in Various Mathematical Texts?

In summary, the Primary Decomposition Theorem states that every module over a commutative ring can be decomposed into a direct sum of primary submodules, which are modules that cannot be further decomposed. This theorem is important in the study of abstract algebra and has applications in many areas, such as algebraic geometry and representation theory. The theorem allows for a better understanding of the structure of modules and their submodules, and plays a crucial role in the classification and characterization of modules.
  • #1
Treadstone 71
275
0
"Let [tex]m(x)[/tex] be the minimal polynomial of [tex]T:V\rightarrow V, \dim V<\infty[/tex] such that [tex]m(x)=m_1(x)m_2(x)[/tex] where [tex]gcd(m_1,m_2)=1[/tex], then there exists [tex]T[/tex]-invariant subspaces [tex]V_1, V_2[/tex] such that [tex]V=V_1\oplus V_2[/tex]."

What other names is this thoerem called? It was given to me as the "primary decomposition theorem" but it's neither in my book nor in mathworld or wikipedia.
 
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  • #2
look on my webpage, http://www.math.uga.edu/~roy/ and download my linear algebra notes. this is proved there.

the proof is based on the linear combination proeprty for gcd's, i.e. you can write 1 as a linear combination of two polynbomilas with gcd = 1, so if 1 = pm1 + qm2, then plugging in T for X, we get id = p(T)m1q(T) +q(T)m2(T). so we have decomposed the identity map into two direct sum components. this decompsoes V accordingly. i.e. V1 = ker pm1, and V2 = ker qm2. is that right? (i am a little under the weather at the moment.)
 
  • #3
Yup, this is it. Thanks.
 
  • #4
my pleasure. we live in the hope of being of service, and occsionally this occurs.
 

Related to How Is the Primary Decomposition Theorem Known in Various Mathematical Texts?

What is the Primary Decomposition Theorem?

The Primary Decomposition Theorem is a fundamental theorem in commutative algebra that states that any finitely generated module over a Noetherian ring can be decomposed into a direct sum of primary modules.

What does the Primary Decomposition Theorem tell us about modules?

The Primary Decomposition Theorem tells us that any module can be broken down into simpler modules called primary modules, which have specific properties that make them easier to analyze and understand.

What is the significance of the Primary Decomposition Theorem?

The Primary Decomposition Theorem is significant because it allows us to study and understand modules by breaking them down into smaller, more manageable pieces. This makes it a powerful tool in the study of commutative algebra and related fields.

Is the Primary Decomposition Theorem applicable to all modules?

Yes, the Primary Decomposition Theorem is applicable to all finitely generated modules over a Noetherian ring. This includes a wide range of modules that are commonly studied in mathematics and science.

How is the Primary Decomposition Theorem used in practice?

The Primary Decomposition Theorem is used in various branches of mathematics and science, such as algebraic geometry, commutative algebra, and representation theory. It is also used in applications, such as in coding theory and cryptography, to understand and analyze different types of modules and their properties.

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