Commutative Algebra

Commutative algebra is the study of commutative rings, their ideals, and modules over such rings. Commutative algebra is used in both algebraic geometry and algebraic number theory. Polynomial rings, algebraic integer rings, including the ordinary integers Z, and p-adic integers are all examples of commutative rings.The key technical tool in the local study of schemes is commutative algebra.

Noncommutative algebra is the study of rings that are not always commutative; it covers ring theory, representation theory, and Banach algebra theory.

The study of rings in algebraic number theory and algebraic geometry is known as commutative algebra.

Dedekind rings, which are an important class of commutative rings in algebraic number theory, are the rings of algebraic integers. The concept of a value ring arose from modular arithmetic considerations. The concept of integral extensions and integrally closed domains, as well as the concept of ramification of an extension of valuation rings, arose from the restriction of algebraic field extensions to subrings.

One of the key differences between commutative algebra and the theory of non-commutative rings is the concept of ring localization (in particular, localization with regard to a prime ideal, localization consisting of inverting a single element, and the total quotient ring). It gives rise to a significant class of commutative rings known as local rings, which have just one maximal ideal. The Zariski topology is naturally associated with the set of prime ideals of a commutative ring. All of these concepts are widely utilised in algebraic geometry, and they are the fundamental technical tools for defining scheme theory, Grothendieck’s expansion of algebraic geometry.

Commutative theorem

Commutative law only applies to addition and multiplication operations in mathematics. It does not, however, apply to the other two mathematical operations, subtraction and division. If a and b are any two integers, the addition and multiplication of a and b produce the same result regardless of the position of a and b, according to commutative law. It can be represented symbolically as:

• a+b = b+a

• a×b = b×a

Commutative law asserts that when two numbers are added or multiplied, the resultant value remains the same regardless of the position of the two numbers. Alternatively, the sequence in which we add or multiply any two real numbers has no effect on the outcome.

Commutative algebra example

The ring of integers Z is the most basic example of commutative algebra. The unique factorization theorem and the presence of primes lay the groundwork for concepts like Noetherian rings and primary decomposition.

Other noteworthy examples include

• polynomial rings R[x₁,…,x_n]

• Integers that are p-adic

• Algebraic integer rings.

Connection with algebraic geometry

Commutative algebra has always been a part of algebraic geometry (in the form of polynomial rings and their quotients, which are used to define algebraic varieties). In the late 1950s, however, algebraic varieties were absorbed into Alexander Grothendieck’s scheme concept. Their local objects are affine schemes or prime spectra, which are locally ringed spaces that form an anti equivalent (dual) category to the category of commutative unital rings, thus extending the duality between the category of affine algebraic varieties over a field k and the category of finitely generated reduced k-algebras.

The Zariski topology in the set-theoretic sense is then substituted by a Grothendieck topology Zariski topology. Grothendieck topologies were created with more exotic but geometrically finer and more sensitive examples in mind than the basic Zariski topology, such as the étale topology and the two flat Grothendieck topologies: fppf and fpqc. Other examples, such as the Nisnevich topology, have recently gained prominence.

Conclusion

In mathematics, a commutative law is one of two rules relating to addition and multiplication that are symbolically represented as a + b = b + a and ab = ba. Rearranging the terms or components has no effect on any finite sum or product, according to these principles. 

Commutative algebra is used in both algebraic geometry and algebraic number theory. Polynomial rings, algebraic integer rings, including the ordinary integers Z, and p-adic integers are all examples of commutative rings.

Noncommutative algebra is the study of rings that are not always commutative; it covers ring theory, representation theory, and Banach algebra theory.

Dedekind rings, which are an important class of commutative rings in algebraic number theory, are the rings of algebraic integers. 

One of the key differences between commutative algebra and the theory of non-commutative rings is the concept of ring localization. It gives rise to a significant class of commutative rings known as local rings, which have just one maximal ideal. Commutative law only applies to addition and multiplication operations in mathematics. Commutative law asserts that when two numbers are added or multiplied, the resultant value remains the same regardless of the position of the two numbers. The ring of integers Z is the most basic example of commutative algebra.