Acquire a good knowledge of some methods and fundamental results in the study of the commutative rings and their modules, with particular reference to the study of ring classes of interest for the algebraic theory of numbers and for algebraic geometry.
Curriculum
teacher profile teaching materials
Modules and submodules. Operations between submodules. Annihilator. Homomorphisms and quotient modules. Generators and bases. Free modules. Rank invariance.
Direct sum and direct product. Tensor product of
modules. Universal property. Tensor product of algebras. Exactness of
tensor product. Flat modules. Extension and restriction of scalars. The Theorem of Caylay-Hamilton. Nakayama's Lemma.
2. Ideals
Operations between ideals. Homomorphisms of rings and quotient rings. Prime and primary ideals. Zorn's lemma. Maximal and minimal ideals. Jacobson's radical
and Nilradical. Radical ideals. Reduced rings. The Chinese Remainder Theorem. First
Avoidance Theorem. Fractional ideals of domains. Invertible ideals.
3. Rings and modules of fractions
Multiplicative parts. Saturated multiplicative parts. Rings and modules of fractions.
Extension and contraction of ideals. Prime and primary ideals in rings of fractions. Local rings. Local properties. The ring of formal series on a field.
4. Integral dependence
Integral dependence and integral closure. Stability and transitivity properties
of full dependence. Lying over, Inc and Going up. Krull size of the
integral closure. Notes on the noetherian nature of integral closure.
Valutation rings and their characterizations. Discrete valuation rings. Krull theorem on the integral closure. Dedekind Rings
5. Noetherian and Artinian rings and modules
Chain conditions and equivalent properties. Noetherian rings and modules.
Modules and algebras on Noetherian rings. The Hilbert Basis Theorem. Cohen's Theorem. Primary decomposition of ideals. Uniqueness theorems. Associated primes
and zerodivisors. Artinian rings and modules. Artionian Ring characterization theorem. The Principal Ideal Theorem.
6. Completion of rings.
R. Gilmer, Multiplicative Ideal Theory, Dekker, New York, 1972
Mutuazione: 20410445 AL410 - ALGEBRA COMMUTATIVA in Matematica LM-40 SUPINO PAOLA, TARTARONE FRANCESCA
Programme
1. ModulesModules and submodules. Operations between submodules. Annihilator. Homomorphisms and quotient modules. Generators and bases. Free modules. Rank invariance.
Direct sum and direct product. Tensor product of
modules. Universal property. Tensor product of algebras. Exactness of
tensor product. Flat modules. Extension and restriction of scalars. The Theorem of Caylay-Hamilton. Nakayama's Lemma.
2. Ideals
Operations between ideals. Homomorphisms of rings and quotient rings. Prime and primary ideals. Zorn's lemma. Maximal and minimal ideals. Jacobson's radical
and Nilradical. Radical ideals. Reduced rings. The Chinese Remainder Theorem. First
Avoidance Theorem. Fractional ideals of domains. Invertible ideals.
3. Rings and modules of fractions
Multiplicative parts. Saturated multiplicative parts. Rings and modules of fractions.
Extension and contraction of ideals. Prime and primary ideals in rings of fractions. Local rings. Local properties. The ring of formal series on a field.
4. Integral dependence
Integral dependence and integral closure. Stability and transitivity properties
of full dependence. Lying over, Inc and Going up. Krull size of the
integral closure. Notes on the noetherian nature of integral closure.
Valutation rings and their characterizations. Discrete valuation rings. Krull theorem on the integral closure. Dedekind Rings
5. Noetherian and Artinian rings and modules
Chain conditions and equivalent properties. Noetherian rings and modules.
Modules and algebras on Noetherian rings. The Hilbert Basis Theorem. Cohen's Theorem. Primary decomposition of ideals. Uniqueness theorems. Associated primes
and zerodivisors. Artinian rings and modules. Artionian Ring characterization theorem. The Principal Ideal Theorem.
6. Completion of rings.
Core Documentation
M. F. Atiyah, I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley, 1969.R. Gilmer, Multiplicative Ideal Theory, Dekker, New York, 1972
Type of delivery of the course
Lectures by the teacher with sessions of exercises only. In any case, the indications of the University will be followed regarding the possibility of transmitting the lessons on an online platform (Teams). Seminars on some topics chosen by attending students are required.Type of evaluation
A written test and an oral test will be held during the appeals. The written test (including ongoing evaluations) consists of 5/6 practical/theoretical exercises to be carried out in 2.30/3 hours. There will also be two ongoing tests to exempt the student from the written test. Students will also be evaluated on the basis of the seminars they will carry out during the course. teacher profile teaching materials
Modules and submodules. Operations between submodules. Omomorphisms
and quotient modules. Generators and bases. Free modules. Invariance of rank.
Direct sum and direct product. Tensor product of
modules. Universal property. Tensor product of algebras. Exactness of
tensor product. Flat modules. Extension and restriction of scalars. The Theorem
of Caylay-Hamilton. The Nakayama Lemma.
2. Ideals
Operations between ideals. Homomorphisms of rings and quotient rings. Prime and primary ideals.
Zorn's lemma. Maximal and minimal ideals. Jacobson radical
and Nilradical. Radical ideals. Reduced rings. The Chinese Remeinder Theorem. Prime
Avoidance Theorem. Fractional ideals of domains. Invertible ideals.
3. Rings and fraction modules
Multiplicative parts. Saturated multiplicative parts. Rings and fraction modules.
Extension and contraction of ideals. Prime and primary ideals in fraction rings. Local rings.
Local properties. Ring of formal series on a field.
4. Integral dependence
Integral dependence and integral closure. Properties of stability and transitivity
of integral dependence. Lying over, Inc and Going up. Krull dimension of the
integral closure. Notes on the noetherianity of integral closure. Valuation rings and their characterizations. Discrete valuation rings. The Theorem of
Krull on integralclosure. Dedekind rings
5. Noetherian and Artinian rings and modules.
Chain conditions and equivalent properties. Noetherian and Artininan rings.
Modules and algebras on noetherian rings. The Hilbert Base Theorem. The Cohen Theorem.
Primary decomposition of ideals. Uniqueness theorems. Prime associates
and zerodivisori. Rings and artinian modules. Characterization theorem for Artinian rings
The Principal Ideal Theorem.
R. Gilmer, Multiplicative Ideal Theory, Dekker, New York, 1972
Mutuazione: 20410445 AL410 - ALGEBRA COMMUTATIVA in Matematica LM-40 SUPINO PAOLA, TARTARONE FRANCESCA
Programme
1. ModulesModules and submodules. Operations between submodules. Omomorphisms
and quotient modules. Generators and bases. Free modules. Invariance of rank.
Direct sum and direct product. Tensor product of
modules. Universal property. Tensor product of algebras. Exactness of
tensor product. Flat modules. Extension and restriction of scalars. The Theorem
of Caylay-Hamilton. The Nakayama Lemma.
2. Ideals
Operations between ideals. Homomorphisms of rings and quotient rings. Prime and primary ideals.
Zorn's lemma. Maximal and minimal ideals. Jacobson radical
and Nilradical. Radical ideals. Reduced rings. The Chinese Remeinder Theorem. Prime
Avoidance Theorem. Fractional ideals of domains. Invertible ideals.
3. Rings and fraction modules
Multiplicative parts. Saturated multiplicative parts. Rings and fraction modules.
Extension and contraction of ideals. Prime and primary ideals in fraction rings. Local rings.
Local properties. Ring of formal series on a field.
4. Integral dependence
Integral dependence and integral closure. Properties of stability and transitivity
of integral dependence. Lying over, Inc and Going up. Krull dimension of the
integral closure. Notes on the noetherianity of integral closure. Valuation rings and their characterizations. Discrete valuation rings. The Theorem of
Krull on integralclosure. Dedekind rings
5. Noetherian and Artinian rings and modules.
Chain conditions and equivalent properties. Noetherian and Artininan rings.
Modules and algebras on noetherian rings. The Hilbert Base Theorem. The Cohen Theorem.
Primary decomposition of ideals. Uniqueness theorems. Prime associates
and zerodivisori. Rings and artinian modules. Characterization theorem for Artinian rings
The Principal Ideal Theorem.
Core Documentation
M. F. Atiyah, I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley, 1969.R. Gilmer, Multiplicative Ideal Theory, Dekker, New York, 1972
Type of delivery of the course
Lectures by the teacher with sessions of exercises only. In any case, the instructions of the University regarding the possibility of transmitting the lessons on Microsoft Teams will be followed if this becomes necessary for the Covid emergency. It is required that students will discuss some chosen topics as a seminar.Type of evaluation
A written and oral exam are scheduled. The written test (including the partial tests) consists of 5/6 practical / theoretical exercises to be carried out in 2.30 / 3 hours. There will also be two partial tests to exempt the student from the written test. Students will also be assessed on the basis of the seminars they will hold during the course. teacher profile teaching materials
Modules and submodules. Operations between submodules. Annihilator. Homomorphisms and quotient modules. Generators and bases. Free modules. Rank invariance.
Direct sum and direct product. Tensor product of
modules. Universal property. Tensor product of algebras. Exactness of
tensor product. Flat modules. Extension and restriction of scalars. The Theorem of Caylay-Hamilton. Nakayama's Lemma.
2. Ideals
Operations between ideals. Homomorphisms of rings and quotient rings. Prime and primary ideals. Zorn's lemma. Maximal and minimal ideals. Jacobson's radical
and Nilradical. Radical ideals. Reduced rings. The Chinese Remainder Theorem. First
Avoidance Theorem. Fractional ideals of domains. Invertible ideals.
3. Rings and modules of fractions
Multiplicative parts. Saturated multiplicative parts. Rings and modules of fractions.
Extension and contraction of ideals. Prime and primary ideals in rings of fractions. Local rings. Local properties. The ring of formal series on a field.
4. Integral dependence
Integral dependence and integral closure. Stability and transitivity properties
of full dependence. Lying over, Inc and Going up. Krull size of the
integral closure. Notes on the noetherian nature of integral closure.
Valutation rings and their characterizations. Discrete valuation rings. Krull theorem on the integral closure. Dedekind Rings
5. Noetherian and Artinian rings and modules
Chain conditions and equivalent properties. Noetherian rings and modules.
Modules and algebras on Noetherian rings. The Hilbert Basis Theorem. Cohen's Theorem. Primary decomposition of ideals. Uniqueness theorems. Associated primes
and zerodivisors. Artinian rings and modules. Artionian Ring characterization theorem. The Principal Ideal Theorem.
6. Completion of rings.
R. Gilmer, Multiplicative Ideal Theory, Dekker, New York, 1972
Mutuazione: 20410445 AL410 - ALGEBRA COMMUTATIVA in Matematica LM-40 SUPINO PAOLA, TARTARONE FRANCESCA
Programme
1. ModulesModules and submodules. Operations between submodules. Annihilator. Homomorphisms and quotient modules. Generators and bases. Free modules. Rank invariance.
Direct sum and direct product. Tensor product of
modules. Universal property. Tensor product of algebras. Exactness of
tensor product. Flat modules. Extension and restriction of scalars. The Theorem of Caylay-Hamilton. Nakayama's Lemma.
2. Ideals
Operations between ideals. Homomorphisms of rings and quotient rings. Prime and primary ideals. Zorn's lemma. Maximal and minimal ideals. Jacobson's radical
and Nilradical. Radical ideals. Reduced rings. The Chinese Remainder Theorem. First
Avoidance Theorem. Fractional ideals of domains. Invertible ideals.
3. Rings and modules of fractions
Multiplicative parts. Saturated multiplicative parts. Rings and modules of fractions.
Extension and contraction of ideals. Prime and primary ideals in rings of fractions. Local rings. Local properties. The ring of formal series on a field.
4. Integral dependence
Integral dependence and integral closure. Stability and transitivity properties
of full dependence. Lying over, Inc and Going up. Krull size of the
integral closure. Notes on the noetherian nature of integral closure.
Valutation rings and their characterizations. Discrete valuation rings. Krull theorem on the integral closure. Dedekind Rings
5. Noetherian and Artinian rings and modules
Chain conditions and equivalent properties. Noetherian rings and modules.
Modules and algebras on Noetherian rings. The Hilbert Basis Theorem. Cohen's Theorem. Primary decomposition of ideals. Uniqueness theorems. Associated primes
and zerodivisors. Artinian rings and modules. Artionian Ring characterization theorem. The Principal Ideal Theorem.
6. Completion of rings.
Core Documentation
M. F. Atiyah, I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley, 1969.R. Gilmer, Multiplicative Ideal Theory, Dekker, New York, 1972
Type of delivery of the course
Lectures by the teacher with sessions of exercises only. In any case, the indications of the University will be followed regarding the possibility of transmitting the lessons on an online platform (Teams). Seminars on some topics chosen by attending students are required.Type of evaluation
A written test and an oral test will be held during the appeals. The written test (including ongoing evaluations) consists of 5/6 practical/theoretical exercises to be carried out in 2.30/3 hours. There will also be two ongoing tests to exempt the student from the written test. Students will also be evaluated on the basis of the seminars they will carry out during the course. teacher profile teaching materials
Modules and submodules. Operations between submodules. Omomorphisms
and quotient modules. Generators and bases. Free modules. Invariance of rank.
Direct sum and direct product. Tensor product of
modules. Universal property. Tensor product of algebras. Exactness of
tensor product. Flat modules. Extension and restriction of scalars. The Theorem
of Caylay-Hamilton. The Nakayama Lemma.
2. Ideals
Operations between ideals. Homomorphisms of rings and quotient rings. Prime and primary ideals.
Zorn's lemma. Maximal and minimal ideals. Jacobson radical
and Nilradical. Radical ideals. Reduced rings. The Chinese Remeinder Theorem. Prime
Avoidance Theorem. Fractional ideals of domains. Invertible ideals.
3. Rings and fraction modules
Multiplicative parts. Saturated multiplicative parts. Rings and fraction modules.
Extension and contraction of ideals. Prime and primary ideals in fraction rings. Local rings.
Local properties. Ring of formal series on a field.
4. Integral dependence
Integral dependence and integral closure. Properties of stability and transitivity
of integral dependence. Lying over, Inc and Going up. Krull dimension of the
integral closure. Notes on the noetherianity of integral closure. Valuation rings and their characterizations. Discrete valuation rings. The Theorem of
Krull on integralclosure. Dedekind rings
5. Noetherian and Artinian rings and modules.
Chain conditions and equivalent properties. Noetherian and Artininan rings.
Modules and algebras on noetherian rings. The Hilbert Base Theorem. The Cohen Theorem.
Primary decomposition of ideals. Uniqueness theorems. Prime associates
and zerodivisori. Rings and artinian modules. Characterization theorem for Artinian rings
The Principal Ideal Theorem.
R. Gilmer, Multiplicative Ideal Theory, Dekker, New York, 1972
Mutuazione: 20410445 AL410 - ALGEBRA COMMUTATIVA in Matematica LM-40 SUPINO PAOLA, TARTARONE FRANCESCA
Programme
1. ModulesModules and submodules. Operations between submodules. Omomorphisms
and quotient modules. Generators and bases. Free modules. Invariance of rank.
Direct sum and direct product. Tensor product of
modules. Universal property. Tensor product of algebras. Exactness of
tensor product. Flat modules. Extension and restriction of scalars. The Theorem
of Caylay-Hamilton. The Nakayama Lemma.
2. Ideals
Operations between ideals. Homomorphisms of rings and quotient rings. Prime and primary ideals.
Zorn's lemma. Maximal and minimal ideals. Jacobson radical
and Nilradical. Radical ideals. Reduced rings. The Chinese Remeinder Theorem. Prime
Avoidance Theorem. Fractional ideals of domains. Invertible ideals.
3. Rings and fraction modules
Multiplicative parts. Saturated multiplicative parts. Rings and fraction modules.
Extension and contraction of ideals. Prime and primary ideals in fraction rings. Local rings.
Local properties. Ring of formal series on a field.
4. Integral dependence
Integral dependence and integral closure. Properties of stability and transitivity
of integral dependence. Lying over, Inc and Going up. Krull dimension of the
integral closure. Notes on the noetherianity of integral closure. Valuation rings and their characterizations. Discrete valuation rings. The Theorem of
Krull on integralclosure. Dedekind rings
5. Noetherian and Artinian rings and modules.
Chain conditions and equivalent properties. Noetherian and Artininan rings.
Modules and algebras on noetherian rings. The Hilbert Base Theorem. The Cohen Theorem.
Primary decomposition of ideals. Uniqueness theorems. Prime associates
and zerodivisori. Rings and artinian modules. Characterization theorem for Artinian rings
The Principal Ideal Theorem.
Core Documentation
M. F. Atiyah, I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley, 1969.R. Gilmer, Multiplicative Ideal Theory, Dekker, New York, 1972
Type of delivery of the course
Lectures by the teacher with sessions of exercises only. In any case, the instructions of the University regarding the possibility of transmitting the lessons on Microsoft Teams will be followed if this becomes necessary for the Covid emergency. It is required that students will discuss some chosen topics as a seminar.Type of evaluation
A written and oral exam are scheduled. The written test (including the partial tests) consists of 5/6 practical / theoretical exercises to be carried out in 2.30 / 3 hours. There will also be two partial tests to exempt the student from the written test. Students will also be assessed on the basis of the seminars they will hold during the course.