A maximal ideal of a ring R is an ideal that is not contained in any proper ideal of R. For example, 2ℤ is a maximal ideal of ℤ, but 4ℤ is not a maximal of ℤ as 4ℤ ⊂ 2ℤ. In this article, we will study maximal ideals, its definition, examples with some solved … Read more
The ring Zn of integers modulo n is an integral domain when n is a prime number. In this article, we will prove that Zn is an integral domain if and only if n is a prime number. What is an Integral Domain? A non-trivial ring R is said to be an integral domain if … Read more
The characteristic of a field is the smallest positive integer n such that nI = 0 where I denotes the multiplicative identity. If no such n exists, then the characteristic is zero. It is denoted by the symbol char(F). In this article, the characteristic of a field is either 0 or a prime number. Characteristic … Read more
A field is an integral domain, but the converse is not true. In this post, we will prove this fact. Before we prove, let us recall what are fields and integral domains. Field: A field is a non-trivial commutative ring with unity where each non-zero element is a unit. For example, (ℝ, +, ⋅) is … Read more
Field theory in Mathematics is an important topic where we study sets equipped with two operations + and ×. In this article, we will study fields in abstract algebra with its definition, examples, and properties. Definition A non-trivial commutative ring with unity is called a field if its every non-zero elements is a unit. More … Read more
A commutative non-trivial ring with unity and no zero divisors is called an integral domain. For example, the set Z of integers is an integral domain. In this article, let us study integral domain along with its definition, examples, and a few solved problems. Definition of an Integral Domain Let R be a non-trivial ring … Read more
A finite integral domain is a field. If R is a finite integral domain, then it must be a field. In this article, we will prove that every finite integral domain is a field. Every finite integral domain is field Theorem: Prove that every finite integral domain is a field Proof: Let R be a … Read more
In this article, we will learn about idempotent element and nilpotent element of a ring with examples. Definition of an Idempotent Element Let R be a ring. An element e in R is called an idempotent element if e2 = e. For example, the zero element 0 is an idempotent element as 02 = 0. … Read more
A zero divisor of a ring (R, +, ⋅) is an element such that its product with a non-zero element is equal to 0. In this article, we will learn about the divisors of zero in a ring with examples, and how to find zero divisors. Definition of a Zero Divisor Let (R, +, ⋅) … Read more
A positive integer n is said to be the characteristic of a ring R, if it is the least number such that na = 0 for all a ∈ R. In this article, we will study the characteristic of a ring along with its definition, examples, and theorems. Definition The characteristic of a ring R … Read more