Number Theory Part 3: Integral Domain

In continuation with my previous blog, this blog will discuss the ring theory.

Integral Domain

An integral domain {R, +, ×} is a commutative ring that satisfies the following two additional properties:

M5. Multiplicative inverse: The set R must include an identity element for the multiplicative operation. That is a×1=a×1=a, where a is an element in set R.

M6. No zero divisor: Let 0 denote the identity element for the addition operation. If a multiplication of any two elements a and b of R results in 0, if ab = 0, then either a or b must be 0.

No Zero Divisor: Let’s make it simple. A no-zero divisor is an element in a ring that does not have the property of a zero divisor. In simpler terms, if ab = 0 implies that either a or b (or both) equals zero, then a and b are considered no zero divisors.

Here’s an example to illustrate the concept:

1. Consider the ring R = ℤ₆, which consists of the integers modulo 6. The elements of this ring are {0, 1, 2, 3, 4, 5}. Multiplication is performed modulo 6. In this ring, the element 2 is a zero divisor because whene 2 is multiplied with any element in the ring, which is 2 × 3 ≡ 0 (mod 6), neither 2 nor 3 is equal to zero. So, in the ring Z₆, element 3 is a no-zero divisor.
2. Consider another ring R = ℤ₇, which consists of the integers modulo 7. The elements of this ring are {0, 1, 2, 3, 4, 5, 6}. Let’s analyse the element 2 in R. To determine if 2 is a zero divisor; we need to check if there exists a nonzero element a in the ring such that 2×a ≡ 0 (mod 7).

Multiplying 2 by each nonzero element in the ring:

2 ×1 ≡ 2 (mod 7)
2 ×2 ≡ 4 (mod 7)
2 ×3 ≡ 6 (mod 7)
2 ×4 ≡ 1 (mod 7)
2 ×5 ≡ 3 (mod 7)
2 ×6 ≡ 5 (mod 7)

As we can see, there is no nonzero element for which the product with 2 yields zero. Therefore, 2 is a no-zero divisor in Z₇.

I have two questions the reader wants to answer before proceeding to the next stepping stone.

Ques1: Why do no zero divisors play a pivot role in modern cryptography?

Ans: No zero divisors play a significant role in modern public key cryptography due to two main reasons:

1. Invertibility: The security of Public key cryptography relies on multiplicative inverses. In a ring with zero divisors, not all elements have multiplicative inverses, which can lead to vulnerabilities in the cryptographic system. No Zero Divisor ensures that elements have multiplicative inverses, enhancing cryptographic security.
2. Avoiding Information Leakage: Zero divisors in cryptographic algorithms can leak private key or secret parameter information. To mitigate this risk, using rings without zero divisors can strengthen the confidentiality of cryptographic operations.

Ques 2: What are the special numbers in algebra that identify no zero divisors in a ring?

Ans: The answer is prime numbers that play an important role in identifying no-zero divisors in a ring, which are used to construct the multiplicative inverses.

For example, consider the ring of integers modulo 7. This ring is a field, meaning every nonzero element has a multiplicative inverse. The multiplicative inverse of 2 in this ring is 5, because 2×10≡1(mod7).

Therefore, prime numbers are essential for ensuring that there are no zero divisors in a ring. Without prime numbers, it would be impossible to construct the multiplicative inverses, and this would mean there would be zero-divisors in the ring.

In summary, Integral domains are fundamental algebraic structures in abstract algebra. They are commutative rings with no zero divisors, meaning that the product of any two non-zero elements is non-zero. Integral domains have diverse applications in mathematics, including number theory and algebraic geometry. They are also helpful in computer science and cryptography for coding theory, error correction, and cryptographic algorithms. Integral domains are essential in mathematics and play a significant role in solving practical problems in various fields.

Number Theory Part 1: Group

Number Theory Part 2: Ring

Number Theory Part 4: Fields