Abstract
Throughout the manuscript, unless explicitly stated otherwise, rings are commutative and unitary, and a ring homomorphism \(\varphi :\mathbf {A}\rightarrow \mathbf {B}\) must satisfy \(\varphi (1_\mathbf {A})=1_\mathbf {B}\).
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In this introductory chapter, when we use the incantatory figurative expression field \(\mathbf {K}\) “extension of \(\mathbf {R}\),” we simply mean that \(\mathbf {K}\) is a field with an \(\mathbf {R}\)-algebra structure. This boils down to saying that a subring of \(\mathbf {K}\) is isomorphic to a (integral) quotient of \(\mathbf {R}\), and that the isomorphism is given. Consequently the coefficients of \(f\) can be “seen” in \(\mathbf {K}\) and the speech following the incantatory expression does indeed have a precise algebraic meaning. In Chap. III we will define a ring extension as an injective homomorphism. This definition directly conflicts with the figurative expression used here if \(\mathbf {R}\) is not a field. This explains the inverted commas used in the current chapter.
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© 2015 Springer Science+Business Media Dordrecht
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Lombardi, H., Quitté, C. (2015). Examples. In: Commutative Algebra: Constructive Methods. Algebra and Applications, vol 20. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-9944-7_1
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DOI: https://doi.org/10.1007/978-94-017-9944-7_1
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