Polynomials Ring without Unit

Oct 13, 2025 by magicalceylontours.com 106 views

Introduction

In abstract algebra, a zero divisor is an element that, when multiplied by a non-zero element, results in zero. This concept is crucial in understanding the properties of rings, particularly in the context of polynomials. In this article, we will delve into the general definition of zero divisors in a polynomials ring, specifically focusing on a ring without unit. We will explore the properties of this ring and provide examples to illustrate the concept.

Polynomials Ring without Unit

Let $I$ be a set and $A$ a ring without unit. The polynomials ring, denoted as $\left\{P(I,A),+,\cdot\right\}$ , consists of functions $f:^{\,(I)}\mathbb{N}\to A$ with finite support. In other words, each function $f$ is a mapping from the natural numbers to the ring $A$ , with only a finite number of non-zero values.

Definition of Zero Divisors

A zero divisor in the polynomials ring $P(I,A)$ is an element $f$ such that there exists a non-zero element $g$ in $P(I,A)$ with the property that $f \cdot g = 0$ . Here, the multiplication operation $\cdot$ is defined as the pointwise multiplication of functions, i.e., $(f \cdot g)(n) = f(n) \cdot g(n)$ for all $n \in \mathbb{N}$ .

Properties of Zero Divisors

Zero divisors in the polynomials ring $P(I,A)$ exhibit some interesting properties. Firstly, if $f$ is a zero divisor, then there exists a non-zero element $g$ such that $f \cdot g = 0$ . This implies that $f$ is not a unit, as it does not have a multiplicative inverse.

Example 1: Zero Divisors in a Simple Ring

Consider the ring $A = \mathbb{Z}$ , which consists of integers with the usual addition and multiplication operations. Let $I = \{1, 2\}$ be a set with two elements. The polynomials ring $P(I,A)$ consists of functions $f:^{\,(I)}\mathbb{N}\to \mathbb{Z}$ with finite support.

Suppose we have a function $f$ defined as $f(1) = 2$ and $f(2) = 3$ . This function is not a zero divisor, as there is no non-zero function $g$ such that $f \cdot g = 0$ .

However, consider the function $g$ defined as $g(1) = 1$ and $g(2) = -1$ . We have $f \cdot g = 0$ , as $(f \cdot g)(1) = f(1) \cdot g(1) = 2 \cdot 1 = 2$ and $(f \cdot g)(2) = f(2) \cdot g(2) = 3 \cdot (-1) = -3$ . Since $f \cdot g = 0$ , we conclude that $f$ is a zero divisor.

Example 2: Zero Divisors in a More Complex Ring

Consider the ring $A = \mathbb{Z}_6$ , which consists of integers modulo 6 with the usual addition and multiplication operations. Let $I = \{1, 2, 3\}$ be a set with three elements. The polynomials ring $P(I,A)$ consists of functions $f:^{\,(I)}\mathbb{N}\to \mathbb{Z}_6$ with finite support.

Suppose we have a function $f$ defined as $f(1) = 2$ , $f(2) = 3$ , and $f(3) = 4$ . This function is not a zero divisor, as there is no non-zero function $g$ such that $f \cdot g = 0$ .

However, consider the function $g$ defined as $g(1) = 1$ , $g(2) = 2$ , and $g(3) = 3$ . We have $f \cdot g = 0$ , as $(f \cdot g)(1) = f(1) \cdot g(1) = 2 \cdot 1 = 2$ , $(f \cdot g)(2) = f(2) \cdot g(2) = 3 \cdot 2 = 6 \equiv 0 \pmod{6}$ , and $(f \cdot g)(3) = f(3) \cdot g(3) = 4 \cdot 3 = 12 \equiv 0 \pmod{6}$ . Since $f \cdot g = 0$ , we conclude that $f$ is a zero divisor.

Conclusion

In conclusion, zero divisors in the polynomials ring $P(I,A)$ are elements that, when multiplied by a non-zero element, result in zero. These elements exhibit some interesting properties, such as not being units and having a multiplicative inverse. We have provided examples to illustrate the concept of zero divisors in simple and more complex rings.

References

[1] Dummit, D. S., & Foote, R. M. (2004). Abstract algebra. John Wiley & Sons.
[2] Herstein, I. N. (1996). Abstract algebra. Prentice Hall.
[3] Lang, S. (2002). Algebra. Springer-Verlag.

Introduction

In our previous article, we explored the concept of zero divisors in polynomials ring, specifically focusing on a ring without unit. We defined zero divisors as elements that, when multiplied by a non-zero element, result in zero. In this article, we will provide a Q&A guide to help you better understand the concept of zero divisors in polynomials ring.

Q: What is a zero divisor in a polynomials ring?

A: A zero divisor in a polynomials ring is an element that, when multiplied by a non-zero element, results in zero. In other words, if $f$ is a zero divisor, then there exists a non-zero element $g$ such that $f \cdot g = 0$ .

Q: What is the difference between a zero divisor and a unit in a polynomials ring?

A: A unit in a polynomials ring is an element that has a multiplicative inverse. In other words, if $f$ is a unit, then there exists an element $g$ such that $f \cdot g = 1$ . A zero divisor, on the other hand, does not have a multiplicative inverse.

Q: Can a zero divisor be a unit in a polynomials ring?

A: No, a zero divisor cannot be a unit in a polynomials ring. By definition, a zero divisor does not have a multiplicative inverse, which means it cannot be a unit.

Q: What are some examples of zero divisors in a polynomials ring?

A: Here are some examples of zero divisors in a polynomials ring:

Consider the ring $A = \mathbb{Z}$ and the polynomials ring $P(I,A)$ , where $I = \{1, 2\}$ . Suppose we have a function $f$ defined as $f(1) = 2$ and $f(2) = 3$ . This function is a zero divisor, as there exists a non-zero function $g$ such that $f \cdot g = 0$ .
Consider the ring $A = \mathbb{Z}_6$ and the polynomials ring $P(I,A)$ , where $I = \{1, 2, 3\}$ . Suppose we have a function $f$ defined as $f(1) = 2$ , $f(2) = 3$ , and $f(3) = 4$ . This function is a zero divisor, as there exists a non-zero function $g$ such that $f \cdot g = 0$ .

Q: Can a zero divisor be a non-zero element in a polynomials ring?

A: Yes, a zero divisor can be a non-zero element in a polynomials ring. In fact, the definition of a zero divisor does not require the element to be zero.

Q: What are some properties of zero divisors in a polynomials ring?

A: Here are some properties of zero divisors in a polynomials ring:

A zero divisor is not a unit.
A zero divisor does not have a multiplicative inverse.
A zero divisor can be a non-zero element.
A zero divisor can be a unit in a different ring.

Q: How do zero divisors relate to the concept of ideals in a polynomials ring?

A: Zero divisors are closely related to the concept of ideals in a polynomials ring. In fact, a zero divisor is an element that belongs to an ideal. An ideal is a subset of a ring that is closed under addition and multiplication by elements of the ring.

Q: Can a zero divisor be a generator of an ideal in a polynomials ring?

A: Yes, a zero divisor can be a generator of an ideal in a polynomials ring. In fact, a zero divisor is an element that generates an ideal.

Conclusion

In conclusion, zero divisors in polynomials ring are elements that, when multiplied by a non-zero element, result in zero. These elements exhibit some interesting properties, such as not being units and having a multiplicative inverse. We have provided a Q&A guide to help you better understand the concept of zero divisors in polynomials ring.

References

[1] Dummit, D. S., & Foote, R. M. (2004). Abstract algebra. John Wiley & Sons.
[2] Herstein, I. N. (1996). Abstract algebra. Prentice Hall.
[3] Lang, S. (2002). Algebra. Springer-Verlag.