Rings and Modules

The Jacobson radical

HEGL Illustrating Mathematics Seminar
Winter Semester 2024-2025

Katrin Weiß
Heidelberg University

Recall to last talk

Basic Algebraic structures

• Monoids and Groups
• Rings and Fields (recall of the main aspects for this presentation)
• Heyting Field
• Introduction to Euclidean Algorithm

Monoids and Groups (from last talk)

Monoids:

Be a set and :

• (multiplication) .
• (assoziative law)
• (identity) so that:
• (abelian) :

In a muliplicative monoid

Unit:

:

• is an inverse of :
• We also write
• Then is a unit, meaning that it is invertible

Groups:

• "Next step" after monoids
• Every element is a unit
• | is invertible

Structure of the talk

• Definitions of rings, quasi-regular and nilpotent elements

• Theorems and lemmas about ideals

• The Jacobson radical (possible project)

• Nakayama Lemma

Rings

A ring is an additive abelian group which is also a multiplicative monoid

• is a group and is a monoid

• distributive laws:

• trivial if

Quasi-regular elements

• An element of a ring is called left (right) quasi-regular if has a left (right) inverse : ()

• If is both left and right quasi-regular, then we say that is quasi-regular.

Nilpotent elements

• An element in is nilpotent if for .

• has no nilpotent elements if whenever is nilpotent.

Nilpotent elements are quasi-regular

• If and , then , so nilpotent elements are quasi-regular.

• A left ideal of is called quasi-regular if each element of is left quasi-regular.

Theorem 1

Let be a quasi-regular left ideal. Then each element of is quasi-regular.

Proof:
If , then for some .
As , the element has a left inverse t.
Thus , so .

Lemma 2

If is left (right) quasi-regular, then so is .

Proof:

Define .
If , then .
(If , then .)
So we showed that ba is also left (right) quasi-regular.

Theorem 3

The set of all elements such that is a quasi-regular left ideal is a two-sided ideal.

Proof:

In Lemma 2, we know that the set is closed under two-sided multiplication. So it is sufficient to show that if and are quasi-regular, then so is .
Choose such that , and such that .
Then .

Jacobson radical

The two-sided ideal in Theorem 3 is called the Jacobson radical of R.
{ | is quasi-regular}

Alternative (classical) definition of Jacobson Radical

= ∩ { | is maximum left (or right) ideal of }

Proof of the equivalence

1. Any element so that is quasi-regular is contained in every maximal left ideal of .

2. Any element contained in all maximal left ideals generates a quasi-regular left ideal.

→ possible project

Constructive advantages using the definition of quasi-regular elements

1. Easier to prove if an element has an invert
2. Definition does not require a ring to be commutative → definition is more flexible
3. Clear and understandable theory → properties and behaviour of the ring linked to a clear condition

→ Easier to implement and more efficient for calculating

To dos for the project

• Overthink both definitions of the Jacobson radical
• Develop a constructive proof of the equivalence

Nakayama Lemma

Let be a finitely generated left -module and a quasi-regular left ideal of .
If , then .

Proof:

Assume that is the Jacobson radical of , a two-sided ideal.
Let generate .
Then , so we can write where each is in , as is a two-sided ideal. Thus , so is generated by , and we are done by "induction" on n.

Rings and Modules

HEGL Illustrating Mathematics Seminar

Winter Semester 2024-2025

Johannes Kadel

Heidelberg University

Modules and Submodules

• Module: An -module is an abelian group with a scalar multiplication by elements of a ring satisfying:

  • for all , .
  • for all , .
  • for all , .
  • if has a multiplicative identity.

• Submodule: A subset that is itself an -module under the operations inherited from .

• Vector spaces with scalar multiplication by elements of a ring instead of a field

Examples of Modules

• Vector Spaces: Modules over a field .

• Abelian Groups: Modules over .

• Polynomial Rings: is a module over itself.

Finitely Generated Modules

• Definition: An -module is finitely generated if there exist such that every element can be expressed as:

  for some .

• Examples:

  • Vector space over field of dimension , Generator: any basis
  • , Generator: the element
  • is finitely generated as a -module.

Noetherian Modules

• An -module is Noetherian if every ascending chain of submodules stabilizes.

• Ascending Chain Condition (ACC):

  For every sequence of submodules:

  there exists such that .

Noetherian Rings

• Ring is Noetherian if is Noetherian as a module over itself.
• -> Noetherian for each sequence of finitely generated ideal of ther is such that .

Examples

• Ring of integers

  • Every ideal in is of the form for some integer .

• Polynomial Ring is Noetherian

Theorem 2.1

Let be a submodule of an -module . Then is Noetherian if and only if both and are Noetherian.

Proof of Theorem 2.1 (⇒ Direction)

• Assume is Noetherian.

• is chain of finitely generated submodules of

• Any submodule of , including , is finitely generated.

• We can construct a chain of finitely generated submodules of such that

• such that

Proof of Theorem 2.1 (⇐ Direction)

• Assume and are Noetherian.

• Let be a chain in .

• Consider the chain in :

• Since is Noetherian, such that for .

• We can write and since we can write where submodule of .

• Since is Noetherian, chain stabilizes.

Finite Rank Free Modules

• Definition: A module is free if it has a basis; i.e., elements such that every can be uniquely written:

• Finite Rank: The basis is finite.

• Example: is free of rank .

Finitely Presented Modules

• Present algebraic structures by specifying set of generators and set of relations

• Example: Abelian Group

  • generators:
  • relations: and

Finitely Presented Modules

• Definition: -module is finitely presented if there is a map from a finite-rank free -module onto such that the kernel of is finitely generated.

Theorem 5.1

Statement:

If is a finitely presented -module, and is a map from onto , then the kernel of is finitely generated.

Theorem 5.1

• is a finitely presented -module.
  • There exists a finite-rank free module and a surjective homomorphism with , which is finitely generated.
  • There exists another surjective homomorphism with .

Construct maps

• such that .
• such that .

We do this by defining and on the finite bases of and .

Map to by taking ( ) to ( ) where

and map to by setting

Rings and Modules

HEGL Illustrating Mathematics Seminar

Winter Semester 2024-2025

Johannes Kadel

Heidelberg University

Modules and Submodules

• Module: An -module is an abelian group with a scalar multiplication by elements of a ring satisfying:

  • for all , .
  • for all , .
  • for all , .
  • if has a multiplicative identity.

• Submodule: A subset that is itself an -module under the operations inherited from .

• Vector spaces with scalar multiplication by elements of a ring instead of a field

Examples of Modules

• Vector Spaces: Modules over a field .

• Abelian Groups: Modules over .

• Polynomial Rings: is a module over itself.

Finitely Generated Modules

• Definition: An -module is finitely generated if there exist such that every element can be expressed as:

  for some .

• Examples:

  • Vector space over field of dimension , Generator: any basis
  • , Generator: the element
  • is finitely generated as a -module.

Noetherian Modules

• An -module is Noetherian if every ascending chain of submodules stabilizes.

• Ascending Chain Condition (ACC):

  For every sequence of submodules:

  there exists such that .

Noetherian Rings

• Ring is Noetherian if is Noetherian as a module over itself.
• -> Noetherian for each sequence of finitely generated ideal of ther is such that .

Examples

• Ring of integers

  • Every ideal in is of the form for some integer .

• Polynomial Ring is Noetherian

Theorem 2.1

Let be a submodule of an -module . Then is Noetherian if and only if both and are Noetherian.

Proof of Theorem 2.1 (⇒ Direction)

• Assume is Noetherian.

• is chain of finitely generated submodules of

• Any submodule of , including , is finitely generated.

• We can construct a chain of finitely generated submodules of such that

• such that

Proof of Theorem 2.1 (⇐ Direction)

• Assume and are Noetherian.

• Let be a chain in .

• Consider the chain in :

• Since is Noetherian, such that for .

• We can write and since we can write where submodule of .

• Since is Noetherian, chain stabilizes.

Finite Rank Free Modules

• Definition: A module is free if it has a basis; i.e., elements such that every can be uniquely written:

• Finite Rank: The basis is finite.

• Example: is free of rank .

Finitely Presented Modules

• Present algebraic structures by specifying set of generators and set of relations

• Example: Abelian Group

  • generators:
  • relations: and

Finitely Presented Modules

• Definition: -module is finitely presented if there is a map from a finite-rank free -module onto such that the kernel of is finitely generated.

Theorem 5.1

Statement:

If is a finitely presented -module, and is a map from onto , then the kernel of is finitely generated.

Theorem 5.1

• is a finitely presented -module.
  • There exists a finite-rank free module and a surjective homomorphism with , which is finitely generated.
  • There exists another surjective homomorphism with .

Construct maps

• such that .
• such that .

We do this by defining and on the finite bases of and .

Map to by taking ( ) to ( ) where

and map to by setting