AAR Basic definitions and motivations

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Section 3.1 Basic definitions and motivations

Like most algebraic objects, certain adjectives can be applied to rings to refine their properties. Some familiar examples:

Non-commutative rings: \(n\times n\) matrices \((n\ge 2)\text{,}\) Hamilton’s quaternions: \(\mathbb H\text{,}\) the four-dimensional vector space of \(\R\) with basis \(\{1, i, j, k\}\) subject to the relations \(ij = k = -ji\) and \(i^2=j^2=k^2=-1.\)
Commutative rings: fields, \(\Z\text{,}\) \(\Z/n\Z\text{,}\) polynomial rings with coefficients in a commutative ring.
Integral domains: (also called entire rings) are commutative rings with identity and no zero divisors such as \(\Z\text{,}\) \(\Z/n\Z\) (iff \(n\) is prime), polynomial rings whose coefficient ring is an integral domain.

Much of ring theory evolved to accommodate generalizations of properties of the integers to more general rings. Attempts to extend the notion of unique factorization in the integers led both to the notion and importance of ideals and to entire subjects like algebraic number theory and algebraic geometry.