Ring Theory -- Notation Guide

• - a ring with addition and multiplication
  • - sum of ring elements
  • - product of ring elements
• - a generic ring.
• - a generic ring element
• - additive identity
• - multiplicative identity
• - additive inverse of
• - times. Assuming . The context will be clear if we mean multiplication or addition of multiple summands

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• - a homomorphism
• - kernel of homomorphism .
• - is isomorphic to
• - is a sub(ring / field / domain) of .
• - order of a ring
• - characteristic of a ring

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• - a generic field.
• - is an extension field of ; is a subfield of .
• - a generic integral domain
• - the ring of quotients over the integral domain .
• - partial ring of quotients for where is closed under multiplication and contains neither nor divisors of
• - quotient over field of .
• - the group ring / algebra over .
• - the set (actually more precisely group) consisting of the units of .

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• - polynomial ring on with indeterminate
• - polynomial ring on with indeterminates .
• - the degree of a polynomial .
• - the evaluation (homomorphism) at
• - the algebraic variety of subset .
• - the leading term of when is written in term order ¹
• - the power product of the leading term of
• - the irreducible polynomial for over .
• - the degree of over .
• - The smallest extension field containing and .
• - the degree of a finite extension over .
• - the algebraic closure of in

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• - the factor ring of by
• - sum of two ideals
• - product of two ideals
• - quotient of two ideals
• - the principal ideal generated by
• - ideal generated by .
• - the radical of a ring based on

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• - the Galois Field of order .

Footnotes §

1. Term order pertains to an ordering where indeterminates are sorted in lexicographic order and each degree is increasing. i.e. ↩