In ring theory, a branch of mathematics, the radical of an ideal is a kind of completion of the ideal. There are several special radicals associated with the entire ring - such as the nilradical and the Jacobson radical, which isolate certain "bad" properties of the ring. A radical ideal is an ideal that is its own radical (this can be phrased as being a fixed point of an operation on ideals called 'radicalization'). In mathematics, ring theory is the study of rings, algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers. ... Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... Radical is derived from the Latin word radix, which means root. In various fields of endeavor, it can mean: Sciences in chemistry, either an atom or molecule with at least one unpaired electron, or a group of atoms, charged or uncharged, that act as a single entity in reaction. ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. ... In mathematics, a fixed point of a function f is an argument x such that f(x) = x; see fixed point (mathematics). ...

Definition

The radical of an ideal I in a commutative ring R, denoted by Rad(I) or √I, is defined as In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ...

Intuitively, one can think of the radical of I as obtained by taking all the possible roots of elements of I. Rad(I) turns out to be an ideal itself, containing I. An ideal that is equal to its radical is called a radical ideal or said to be radical.

Examples

Consider the ring Z of integers. The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ...

1. The radical of the ideal 4Z of integers multiple of 4 is 2Z.
2. The radical of 5Z is 5Z.
3. The radical of 12Z is 6Z.

Proof that the radical is an ideal

Let a and b be in the radical of an ideal I. Then, for some positive integers m and n, aⁿ and b^m are in I. We will show that a + b is in I. Use the binomial theorem to expand (a+b)^n+m−1: In mathematics, the binomial theorem is an important formula giving the expansion of powers of sums. ...

For each i, exactly one of the following conditions will hold:

• i ≥ n
• n+m-1-i ≥ m.

This says that in each expression aⁱb^n+m-1-i, either the exponent of a will be large enough to make this power of a be in I, or the exponent of b will be large enough to make this power of b be in I. Since the product of an element in I with an element in R is in I (as I is an ideal), this product expression will be in I, and then (a+b)^n+m−1 is in I, therefore a+b is in the radical of I.

To finish checking that the radical is an ideal, we take an element a in the radical, with aⁿ in I and an arbitrary element r∈R. Then, (ra)ⁿ = rⁿaⁿ is in I, so ra is in the radical. Thus the radical is an ideal.

The nilradical of a ring

Consider the set of all nilpotent elements of R, which will be called the nilradical of R (and will be denoted by N(R)). As can be easily seen, the nilradical of R is just the radical of the zero ideal (0). This brings about an alternative definition for the (general) radical of an ideal I in R. Define Rad(I) as the preimage of N(R/I), the nilradical of R/I, under the projection map R→R/I. In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n such that xn = 0. ...

To see that the two definitions for the radical of I are equivalent, note first that if r is in the preimage of √(R/I), then for some n, rⁿ is zero in R/I, and hence rⁿ is in I. Second, if rⁿ is in I for some n, then the image of rⁿ in R/I is zero, and hence rⁿ is in the preimage of √(R/I). This alternative definition can be very useful, as we shall see right below. See #Properties below for another characterization of the nilradical.

Jacobson radicals

Main article: Jacobson radical

Let R be any ring, not necessarily commutative. The Jacobson radical of R is the intersection of the annihilators of all simple right R-modules. In ring theory, a branch of abstract algebra, the Jacobson radical of a ring R is an ideal of R which contains those elements of R which in a sense are close to zero. It is denoted by J(R) and can be defined in the following equivalent ways: the... In abstract algebra, a (left or right) module S over a ring R is called simple if it is not the zero module and if its only submodules are 0 and S. Understanding the simple modules over a ring is usually helpful because they form the building blocks of all...

There are several equivalent characterizations of the Jacobson radical, such as:

• J(R) is the intersection of the regular maximal right (or left) ideals of R.
• J(R) is the intersection of all the right (or left) primitive ideals of R.
• J(R) is the maximal right (or left) quasi-regular right (resp. left) ideal of R.

As with the nilradical, we can extend this definition to arbitrary two-sided ideals I by defining J(I) to be the preimage of J(R/I) under the projection map R→R/I.

If R is commutative, the Jacobson radical always contains the nilradical. If the ring R is a finitely generated Z-algebra, then the nilradical is equal to the Jacobson radical, and more generally: the radical of any ideal I will always be equal to the intersection of all the maximal ideals of R that contain I. This says that R is a Jacobson ring.

Properties

• If P is a prime ideal, then R/P is an integral domain, so it cannot have zero divisors, and in particular it cannot have nonzero nilpotents. Hence, the nilradical of R/P is {0}, and its preimage, being P, is a radical ideal.

• By using localization, we can see that Rad(I) is the intersection of all the prime ideals of R that contain I: Every prime ideal is radical, so the intersection J of the prime ideals containing I contains Rad(I). If r is an element of R which is not in Rad(I), then we let S be the set {rⁿ|n is a nonnegative integer}. S is multiplicatively closed, so we can form the localization S^-1R. Form the quotient S^-1R/S^-1I. By Zorn's lemma we can choose a maximal ideal P in this ring. The preimage of P under the maps R→S^-1R→S^-1R/S^-1I is a prime ideal which contains I and does not meet S; in particular, it does not meet r, so r is not in J.

• In particular, the nilradical is equal to the intersection of all prime ideals containing the 0 ideal, but all ideals must contain 0 so the nilradical can alternatively be defined as the intersection of the prime ideals.

In mathematics, a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers. ... In abstract algebra, an integral domain is a commutative ring with 0 ≠ 1 in which the product of any two non-zero elements is always non-zero; that is, there are no zero divisors. ... In abstract algebra, a non-zero element a of a ring R is a left zero divisor if there exists a non-zero b such that ab = 0. ... In abstract algebra, localization is a systematic method of adding multiplicative inverses to a ring. ... Zorns lemma, also known as the Kuratowski-Zorn lemma, is a theorem of set theory that states: Every partially ordered set in which every chain (i. ...

Applications

The primary motivation in studying radicals is the celebrated Hilbert's Nullstellensatz in commutative algebra. A nice, easy to understand version of this theorem states that for an algebraically closed field k, and for any finitely generated polynomial ideal J of it, one has Hilberts Nullstellensatz (German: theorem of zeros) is a theorem in algebraic geometry that relates varieties and ideals in polynomial rings over algebraically closed fields. ... In abstract algebra, commutative algebra is the field of study of commutative rings, their ideals, modules and algebras. ... In mathematics, a field F is said to be algebraically closed if every polynomial of degree at least 1, with coefficients in F, has a zero (root) in F (i. ...

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