10.19 The Jacobson radical of a ring
We recall that the Jacobson radical \text{rad}(R) of a ring R is the intersection of all maximal ideals of R. If R is local then \text{rad}(R) is the maximal ideal of R.
Lemma 10.19.1. Let R be a ring with Jacobson radical \text{rad}(R). Let I \subset R be an ideal. The following are equivalent
I \subset \text{rad}(R), and
every element of 1 + I is a unit in R.
In this case every element of R which maps to a unit of R/I is a unit.
Proof.
If f \in \text{rad}(R), then f \in \mathfrak m for all maximal ideals \mathfrak m of R. Hence 1 + f \not\in \mathfrak m for all maximal ideals \mathfrak m of R. Thus the closed subset V(1 + f) of \mathop{\mathrm{Spec}}(R) is empty. This implies that 1 + f is a unit, see Lemma 10.17.2.
Conversely, assume that 1 + f is a unit for all f \in I. If \mathfrak m is a maximal ideal and I \not\subset \mathfrak m, then I + \mathfrak m = R. Hence 1 = f + g for some g \in \mathfrak m and f \in I. Then g = 1 + (-f) is not a unit, contradiction.
For the final statement let f \in R map to a unit in R/I. Then we can find g \in R mapping to the multiplicative inverse of f \bmod I. Then fg = 1 \bmod I. Hence fg is a unit of R by (2) which implies that f is a unit.
\square
Lemma 10.19.2. Let \varphi : R \to S be a ring map such that the induced map \mathop{\mathrm{Spec}}(S) \to \mathop{\mathrm{Spec}}(R) is surjective. Then an element x \in R is a unit if and only if \varphi (x) \in S is a unit.
Proof.
If x is a unit, then so is \varphi (x). Conversely, if \varphi (x) is a unit, then \varphi (x) \not\in \mathfrak q for all \mathfrak q \in \mathop{\mathrm{Spec}}(S). Hence x \not\in \varphi ^{-1}(\mathfrak q) = \mathop{\mathrm{Spec}}(\varphi )(\mathfrak q) for all \mathfrak q \in \mathop{\mathrm{Spec}}(S). Since \mathop{\mathrm{Spec}}(\varphi ) is surjective we conclude that x is a unit by part (17) of Lemma 10.17.2.
\square
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Comment #3650 by Brian Conrad on
Comment #3746 by Johan on