Problem 1 - Algebra

Question

Es sei \(R\) ein kommutativer Ring mit 1. Begründen Sie, dass die Menge \(R[[X]]:=\left\\{P \mid P: \mathbb{N}_{0} \rightarrow R\right\\}\) mit den Verknüpfungen \(+\) und \(\cdot\), die für \(P, Q \in R[[X]]\) wie folgt erklärt sind: $$ (P+Q)(m):=P(m)+Q(m),(P Q)(m):=\sum_{i+j=m} P(i) Q(j) $$ ein kommutativer Erweiterungsring mit 1 von \(R[X]\) ist - der Ring der formalen Potenzreihen oder kürzer Potenzreihenring über \(R\). Wir schreiben \(P=\sum_{i \in \mathbb{N}_{0}} a_{i} X^{i}\) oder \(\sum_{i=0}^{\infty} a_{i} X^{i}\) (also \(P(i)=a_{i}\) ) für \(P \in R[[X]]\) und nennen die Elemente aus \(R[[X]]\) Potenzreihen. Begründen Sie außerdem: (a) \(R[[X]]\) ist genau dann ein Integritätsbereich, wenn \(R\) ein Integritätsbereich ist. (b) Eine Potenzreihe \(P=\sum_{i \in \mathbb{N}_{0}} a_{i} X^{i} \in R[[X]]\) ist genau dann invertierbar, wenn \(a_{0} \in R^{\times}\)gilt, d.h. \(R[[X]]^{\times}=\left\\{\sum_{i=0}^{\infty} a_{i} X^{i} \mid a_{0} \in R^{\times}\right\\}\) (c) Bestimmen Sie in \(R[[X]]\) das Inverse von \(1-X\) und \(1-X^{2}\).

Step-by-Step Solution

Answer

R[[X]] forms a ring extension of R[X]. R[[X]] is an integral domain if R is. A series is invertible if its constant term is invertible. Inverses: \( (1-X)^{-1} = \sum_{i=0}^{\infty} X^i \), \( (1-X^2)^{-1} = \sum_{i=0}^{\infty} X^{2i} \).

Step 1: Understanding Formal Power Series

Formal power series \( P(X) = \sum_{i=0}^{\infty} a_i X^i \) is an infinite series where coefficients \( a_i \) are elements of a commutative ring \( R \). This series is part of \( R[[X]] \), defined as the set of functions from non-negative integers to \( R \). Operations \( + \) and \( \cdot \) are defined pointwise addition and convolution product respectively.

Step 2: Commutative Ring with Unity

We need to show that \( R[[X]] \) forms a commutative ring with unity. Given any two series \( P, Q \in R[[X]] \), addition and multiplication defined as \((P+Q)(m)=P(m)+Q(m)\) and \((PQ)(m)=\sum_{i+j=m} P(i)Q(j) \) satisfy commutativity, associativity, and distributivity properties. The constant series with all coefficients zero except \( a_0 = 1 \) serves as the identity, showing \( R[[X]] \) is a ring with unity.

Step 3: Subring Criteria - Ring Extension

The ring \( R[X] \) of polynomials is a subring of \( R[[X]] \), where only finitely many \( a_i eq 0 \). Every polynomial in \( R[X] \) is also a power series in \( R[[X]] \), satisfying the criteria for \( R[[X]] \) being a ring extension.

Step 4: Conditions for Integral Domain

(a) For \( R[[X]] \) to be an integral domain, it must be commutative with no zero divisors. Since the definition extends the property of its coefficients' ring \( R \), if \( R \) lacks zero divisors, \( R[[X]] \) does too. Conversely, any zero divisor in \( R \) creates non-trivial zero divisors in \( R[[X]] \), indicating that \( R[[X]] \) is an integral domain if and only if \( R \) is.

Step 5: Conditions for Invertibility

(b) A power series \( P \in R[[X]] \) is invertible if there exists \( Q \in R[[X]] \) with \( PQ=1 \). This is possible only if the constant term \( a_0 \) of \( P \) is invertible in \( R \), as it ensures the existence of an inverse for constantly maintaining the product \( PQ \) as the unity series.

Step 6: Inverse Calculation for Specific Series

(c) To find inverses, use the geometric series formula and match terms:- For \( 1-X \), the inverse is \( 1 + X + X^2 + X^3 + \cdots = \sum_{i=0}^{\infty} X^i \).- For \( 1-X^2 \), the inverse is \( 1 + X^2 + X^4 + X^6 + \cdots = \sum_{i=0}^{\infty} X^{2i} \). These are derived using the formula for geometric series inverses.

Key Concepts

Commutative Ring

A commutative ring, in mathematics, is a foundational structure equipped with two operations: addition and multiplication. These operations are designed such that:

• Commutativity: Both addition and multiplication follow the commutative law. This means for any elements \(a\) and \(b\) in the ring, \(a + b = b + a\) and \(ab = ba\).
• Associativity: Both operations are associative. Thus, \((a + b) + c = a + (b + c)\) and \((ab)c = a(bc)\).
• Distributivity: Multiplication distributes over addition. That is, \(a(b + c) = ab + ac\).

In the context of the ring \(R[[X]]\), which comprises formal power series, these properties ensure that addition and convolution multiplication of power series are consistent with these laws, making \(R[[X]]\) a commutative ring if \(R\) itself is a commutative ring.

Integral Domain

An integral domain is a type of commutative ring where there are no zero divisors. This means if \(a\) and \(b\) are non-zero elements, their product \(ab\) is also non-zero. For \(R[[X]]\) - the ring of formal power series, to be an integral domain, a crucial condition must be met:

• The coefficient ring \(R\) itself must be an integral domain.
• This implies that if \(R\) has no zero divisors, then \(R[[X]]\) will inherit this property.
• Conversely, if \(R\) does have zero divisors, \(R[[X]]\) cannot be an integral domain because these zero divisors would propagate through the series operations.

Therefore, \(R[[X]]\) mirrors the integral domain property based on the characteristics of \(R\).

Invertibility

The property of invertibility within a ring is about finding an element that can serve as a multiplicative inverse of another. For a power series \(P(X) = \sum_{i=0}^{\infty} a_i X^i\) in the ring \(R[[X]]\), invertibility happens under specific conditions:

• The constant term \(a_0\) of the series must be invertible in \(R\).
• This ensures that the series can interact with another series \(Q(X)\) such that \(P(X)Q(X) = 1\) where 1 represents the series whose constant term is 1 and other coefficients are 0.
• If \(a_0\) is not invertible in \(R\), it is impossible to maintain the necessary product of series to achieve unity.

Hence, the ability for a series to be inverted in \(R[[X]]\) essentially depends on the invertibility of its leading coefficient.

Ring Extension

A ring extension involves expanding a ring by adding elements to create a larger ring. This is often done while maintaining the ring's existing properties, such as commutativity and associativity.

• When discussing \(R[X]\) and \(R[[X]]\), the extension from polynomials to formal power series is evident.
• \(R[X]\), the ring of polynomials, consists of elements where only finitely many coefficients are non-zero.
• When extending to \(R[[X]]\), series can have infinitely many non-zero coefficients – moving from polynomials to series allows us to operate within a broader context while preserving ring structure.
• This extension allows deeper analysis and manipulation of series akin to polynomials but with increased flexibility and applications in analysis and number theory.

Overall, by understanding these extensions, mathematicians can explore more complex algebraic structures while adhering to familiar ring properties.