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Chapter 3: Problem 11

Let \(R\) be a ring with unit element. Using its elements we define \(a\) ring \(R\) by defining \(a \oplus b=a+b+1\), and \(a \cdot b=a b+a+b\) where \(a, b \in R\) and where the addition and multiplication on the right-hand side of these relations are those of \(R\). (a) Prove that \(R\) is a ring under the operations \(\oplus\) and \(\because\) (b) What acts as the 0-element of \(R\) ? (c) What acts as the 1-element of \(R ?\) (d) Prove that \(R\) is isomorphic to \(R\).

Short Answer

Expert verified
(a) Verify commutativity, associativity, and identities. (b) -1; (c) 0; (d) Use isomorphism \( f(a) = a-1 \).

Step by step solution

01

Define the Operations

We are given new operations \( a \oplus b = a + b + 1 \) and \( a \because b = ab + a + b \). First, let's analyze these operations to check if they satisfy the ring axioms.
02

Check for Commutativity

Check if \( \oplus \) is commutative. Since \( a + b + 1 = b + a + 1 \), we see that \( \oplus \) is commutative. For \( \because \):\( ab+a+b = ba+b+a \), hence \( \because \) is also commutative.
03

Check for Associativity

For \( \oplus \): Evaluate \( (a \oplus b) \oplus c = (a + b + 1) + c + 1 = a + b + c + 2\) and \( a \oplus (b \oplus c) = a + (b + c + 1) + 1 = a + b + c + 2\). Thus, \( \oplus \) is associative.For \( \because \): \( (a \because b) \because c = (ab + a + b)(1 + c) + ab + a + b + c \). Similarly, evaluate \( a \because (b \because c) \) to confirm associativity.
04

Check for Identity Element for \( \oplus \)

Find \( e \) such that \( a \oplus e = a \). Solve \( a + e + 1 = a \), which gives \( e = -1 \). Then, the identity for \( \oplus \) is \(-1\).
05

Check for Identity Element for \( \because \)

Find \( e \) such that \( a \because e = a \). Solve \( ae + a + e = a \), which gives \( e = 0 \). Thus, the identity for \( \because \) is \(0\).
06

Establish Isomorphism

Define a function \( f(a) = a-1 \) from \( R \) to itself. Check that \( f(a \oplus b) = f(a) + f(b) \) and \( f(a \because b) = f(a) f(b) \). Also, show that \( f \) is bijective. This confirms \( R \) with \( \oplus, \because \) is isomorphic to \( R \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Ring with Unity
A ring with unity is a special type of ring that includes a multiplicative identity element. This means there exists an element, often denoted as 1, such that for any element \( a \) in the ring, the equation \( a imes 1 = a = 1 imes a \) holds true. Having a unity element in a ring ensures that the ring behaves somewhat like the familiar set of integers under multiplication. In mathematics, the understanding of rings with unity is crucial because these rings can perform "complete" multiplication, maintaining the structure needed for a variety of algebraic applications. Not all rings have a unity element, which can limit the operations that can be performed within such rings. However, when a unity exists, it broadens the applicability of certain operations and structures within the ring.
Commutative Ring
A commutative ring is a ring in which the multiplication operation is commutative. This means for any two elements \( a \) and \( b \) in the ring, the equation \( a imes b = b imes a \) holds true. Commutative rings are fundamental in algebra because they abstract the essential features of familiar number sets, like integers under multiplication and addition. They form the building blocks for more complex mathematical theories, such as algebraic geometry and number theory.
  • Commutativity simplifies many calculations, allowing for greater versatility and flexibility in problem-solving.
  • Commutative rings appear naturally in many mathematical settings, making them a pivotal concept.
Understanding if a ring is commutative is key in determining what types of algebraic structures and theories can be developed further.
Ring Isomorphism
Ring isomorphism is a concept that establishes a structural similarity between two rings. If two rings \( R_1 \) and \( R_2 \) are isomorphic, there exists a bijective function \( f: R_1 \rightarrow R_2 \) such that for any two elements \( a \) and \( b \) in \( R_1 \), the function satisfies two main properties:
  • \( f(a + b) = f(a) + f(b) \)
  • \( f(a imes b) = f(a) imes f(b) \)
These properties show that addition and multiplication in one ring translate directly into the operations in the other ring. An isomorphism indicates a perfect "match" between two rings in terms of their algebraic structure.Isomorphisms are crucial in ring theory because they allow mathematicians to study different rings "as if" they are the same. By finding an isomorphism, any property that holds in one ring instantly holds in the other. This concept allows much of the richness and depth of ring theory to be transferred between seemingly different mathematical systems.
Ring Identity Elements
Identity elements within a ring are critical for maintaining the integrity of its structure. They help preserve the ring operations, especially concerning addition and multiplication.

Additive Identity

The additive identity, often denoted as 0, in a ring \( R \) is an element that satisfies \( a + 0 = a = 0 + a \) for any element \( a \) in \( R \). It acts as a neutral starting point for addition.

Multiplicative Identity

Similarly, a multiplicative identity in a ring with unity is denoted as 1, satisfying \( a \times 1 = a = 1 \times a \). It serves as a base multiplier that doesn鈥檛 alter any other element in the multiplication operation.Establishing identity elements ensures that every ring operation has a clear and distinct result. These elements create a "home base" in mathematical operations, acting like the 0 and 1 we are familiar with in number systems. Understanding identity elements solidifies one鈥檚 grasp of how rings operate in algebraic contexts and helps identify and establish the foundational properties of different algebraic structures.

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Most popular questions from this chapter

Prove that if an ideal \(U\) of a ring \(R\) contains a unit of \(R\) then \(U=R\).

Let \(R\) be a commutative ring with unit element. A nonempty subset \(S\) of \(R\) is called a multiplicative system if: (1) \(0 \in S\). (2) \(s_{1}, s_{2} \in S\) implies that \(8_{1} s_{2} \in S\). Let \(9 \pi\) be the set of all ordered pairs \((r, s)\) where \(r \in R, s \in S .\) In \(9 \pi\) define \((r, s) \sim\left(r^{\prime}, s^{\prime}\right)\) if there exists an element \(s^{\prime \prime} \in S\) such that $$g^{\prime \prime}\left(r 8^{\prime}-8 r^{\prime}\right)=0 .$$ (a) Prove that this defines an equivalence relation on \(9 \pi\). Let the equivalence class of \((r, s)\) be denoted by \([r, 8]\), and let \(R_{S}\) be the set of all the equivalence classes. In \(R_{S}\) define \(\left[r_{1}, s_{1}\right]+\) \(\left[r_{2}, s_{2}\right]=\left[r_{1} s_{2}+r_{2} s_{1}, 8_{1} s_{2}\right]\) and \(\left[r_{1}, s_{1}\right]\left[r_{2}, s_{2}\right]=\left[r_{1} r_{2}, s_{1} s_{2}\right]\) (b) Prove that the addition and multiplication described above are well- defined and that \(R_{S}\) forms a ring under these operations. (c) Can \(R\) be imbedded in \(R_{S}\) ? (d) Prove that the mapping \(\phi: R \rightarrow R_{z}\) defined by \(\phi(a)=[a s, s]\) is a homomorphism of \(R\) into \(R_{S}\) and find the kernel of \(\phi\). (e) Prove that this kernel has no element of \(S\) in it. (f) Prove that every element of the form \(\left[s_{1}, s_{2}\right]\) (where \(s_{1}, s_{2} \in S\) ) in \(R_{S}\) has an inverse in \(R_{S}\).

Let \(D\) be an integral domain, \(a, b \in D .\) Suppose that \(a^{n}=b^{n}\) and \(a^{m}=b^{m}\) for two relatively prime integers \(m\) and \(n\). Prove that \(a=b\).

If \(R\) is an integral domain, and if \(F\) is its field of quotients, prove that any element \(f(x)\) in \(F[x]\) can be written as \(f(x)=\left(f_{0}(x) / a\right)\), where \(f_{0}(x) \in R[x]\) and where \(a \in R\)

Show that the commutative ring \(D\) is an integral domain if and only if for \(a, b, c \in D\) with \(a \neq 0\) the relation \(a b=a c\) implies that \(b=c\).
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