ring and field

[AntonioLao]

Ring and field of numbers are described clearly in John Derbyshire’s book ‘Prime Obsession,’ chapter 17 with the title ‘A Little Algebra.’ A number field is a set of elements (numbers) that can be added, subtracted, multiplied, and divided using the rules of arithmetic. The answers from these four operations must always create elements (numbers) that also belong to the field in question. Fields can be of finite or infinite size. From these defined properties it is logically understood that the set of the natural counting numbers is not a field since subtracting 11 from 7 gives -4, a negative integer whole number which does not belong to the set of counting numbers. Similarly, the set of integers is not a field since dividing 3 by 2 is 3/2 or 1½ which is not an element of the set of integers. However, adding, subtracting, and multiplying integers do give elements (numbers) belonging to the set. In this sense where and when addition, subtraction, and multiplication are always satisfied while division is not always true, the set is called a ring. From these properties and definitions, it is clear that symmetric singular Hadamard matrices are rings. Furthermore, they are also classified as commutative rings since both matrix addition and matrix multiplication satisfy the commutative property of arithmetic.

[SteveA]

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