Exercises in Basic Ring Theory

An element r of a ring R is called a left (right) zero divisor if there is a nonzero s ∈ R : rs = 0(sr = 0);zero divisor if it is left and right divisor;left (right) cancellable if for every a, b ∈ R : ra = rb(ar = br) ⇒ a = b;idempotent if r2 = r; two idempotents e, e’ are orthogonal if ee’ = e’e = 0; in a ring R we denote by Id(R) the set of all the idempotent elements. A ring is called Boole if all its elements are idempotent.

A subgroup I of (R, +) is called left (right) ideal if R·I = {ri|r ∈ R, i ∈ I} ⊆ I (resp. I·R ⊆ I) and ideal if it is left and right ideal. We denote by (X) = ∩ {I (left or right) ideal in R|X ⊆ I}, if X ⊆ R, called the (left or right) ideal generated by X. In an arbitrary ring $$ \left( X \right) = \left\{ {\sum\limits_{i = 1}^n {{r_i}{x_i} + \sum\limits_{k = 1}^m {{{x'}_k}{{r'}_k}} } + \sum\limits_{s = 1}^l {{{r''}_s}{{x''}_s}} {{r'''}_s} + \sum\limits_{j = 1}^t {{n_j}{y_j}}}\right\}$$with $$ {r_i},{r'_{k,}}{r''_s},{r'''_s} \in R,{x_i},{x'_k},{x''_s},{y_j} \in X,{n_j} \in $$ and the reader can provide the simplier forms if the ring has identity or is commutative (or both).

Zero divisors are defined in the introduction of the first chapter.

Let f : R → R’ a ring homomorphism. If the rings have identity f is called unital if f(1) = 1’.

Denote by ord(a) = ord_(R,+)(a) the (group) order of an element a in a ring R. If there is a m ∈ ℕ* such that ma = 0, ∀a ∈ R then we denote by char(R) the smallest positive integer (if it exists) having this property (i.e. all elements are of finite order and ord(a) divides char(R)). In the remaining case (i.e. there are elements of infinite order or {ord(a)|a ∈ R} is not bounded) we say that char(R) = 0.

If a, b ∈ R a commutative ring, we say that b divides a (denoted b|a) if there is an element c ∈ R: a = bc. One checks b|a ⇔ aR = (a) ⊆ (b) = bR. Two elements are associated (in divisibility) denoted by a ~ b if b|a and a|b. This is an equivalence relation and a ~ b ⇔ aR = bR An element p is calledirreducible if it is not a unit and has no other divisors then units and associated elements;reducible if it is not irreducible;prime if it is not a unit and p|a.b ⇒ p|a or p|b;greatest common divisor for r₁, r₂,…, r _n if p|r _i (1 ≤ i ≤ n) and if d|r_i(1 ≤ i ≤ n) then d|p;least common multiple is defined similarly;Each prime element is irreducible.

A ring K with identity is called a division ring if (K*, ·) is a group. A commutative division ring is called a field.

A ring homomorphism f : R → R’ is called automorphism if it is bijective (an isomorphism) and R = R’ (an endomorphism).

Let M _R be a right R-module, _R N a left R-module and G an abelian group. A ℤ-bilinear map φ : M × N → G is called R-balanced if φ(mr,n) = φ(m,rn) holds for each m ∈ M, n ∈ N, r ∈ R.

A ring R is called left (right) artinian if the set of all the left (right) ideals of R satisfies the dcc (descending chain condition) i.e. each strictly descending chain of left (right) ideals is finite (or equivalently, each non-void family of left (right) ideals contains a minimal element). Similarly, a ring is called left (right) noetherian if the set of all the left (right) ideals satisfies the acc (ascending chain condition).

The left (right) socle of a ring R is the sum of all the minimal (simple) left (right) ideals of R. For commutative rings we shall denote this by s(R).

A ring R is called left (right) semisimple if it is a direct sum of minimal left (right) ideals. A ring is left (right) semisimple iffit coincides with its corresponding socle.

A proper ideal P of a ring R is called prime if for each two ideals I, J in R the inclusion I ° J ⊆ P implies I ⊆ P or J ⊆ P; semiprime if it is an intersection of prime ideals.In a ring with identity every maximal ideal is prime.

For an arbitrary ring with identity R, we denote by R[X] the ring of polynomials of indeterminate X over R.

In this chapter we deal only with commutative rings with identity.

Let X be a nonvoid set. A topology on X is a family τ⊂ P(X) of subsets which is closed under finite intersections and arbitrary unions. The pair (X,τ) is called a topological space, the elements ofτare called the open sets of the space. If τ₁ and τ₂ are topologies on X and τ₁ ⊂ τ₂ we say that τ₂is finer than τ₁ or that τ₁ is coarser than τ₂. It exists in X a finest topology τ° = P(X) (called the discrete topology) and a coarsest one τ₀ = {θ, X} (called the indiscrete topology). The elements of F={X \ G|G ∈ τ} are called the closed sets.

An idempotent element e ≠ 0 is called primitive if it cannot be written as the sum of two orthogonal idempotent elements.

One has only to compute in two different ways (using the distributivity laws) the element: (1 + 1)(a + b) = 1(a + b) + 1(a + b) = a + b + a + b and (1 + 1)(a + b) = (1 + 1)a + (1 + 1) b = a + a + b + b. Now a + b = b + a follows by additive cancellation.

Only P and A are subrings. A is also an ideal but P is not: if f ∈ P and g ∈ F,$$ g(x) = \left\{ {_{0\;\;\;\;\;\;\;\;\;\;\;if\;x\; \ne \;0}^{x\sin \frac{1}{x}\quad if\;x\; \ne \;0}} \right. $$ then f · g has an infinite number of zero’s and hence f · g ∈ P.

If e ∈ R is an idempotent element such that e ∉ {0,1} then obviously e.(1 - e) = 0.

If a’ is arbitrary in f(R) there is an element a ∈ R such that a’ = f(a). Then a’.f(1) = f(a).f(1) = f(a.1) = f(a) = a’ and similarly f(1).a’ = a’ so that f(1) is the identity in f(R). An easy (but not trivial!) counterexample is f : ℤ₁₂ → ℤ₁₂, $$ f(\bar x) = \overline {4x} ,\forall \bar x \in {_{12}} $$. Here 4̄ is the identity in $$f({_{12}}) = \{ \bar 0,\bar 4,\bar 8\} $$ and surely 1̄≠4̄

If a ∈ R, a Boole ring, a + a = (a + a)2 = a + a + a + a so that a + a. = 0. Hence the required characteristic is 2.

If a ≠ 0 is an element of D them by hypothesis a and a2 must be associated elements of D. Then a2 = au for a suitable unit u ∈ D. Hence a(a - u) =0 and a - u = 0, D having no zero divisors. So, each nonzero element in D is a unit.

Let K = {0, 1,a, b} be a field with four elements. Its characteristic being 2 (a prime number dividing 4) we obtain 1 + 1 = a + a = b + b = 0. The group (K, +) being of Klein type we also have a + b = 1. The inveerse of a must be b. Indeed, a-1 = 1 would imply a = 1, a-1 = a would imply a2 = 1 and (a - 1)2 = 0 (char(K) = 2) and hence a = 1. So a-1 = b or ab = 1 and even a(1 + a) = 1 or a2 =1 + a. An isomorphism to K₄ (from 1.9) is now obviously defined

If f : ℚ → ℚ is an automorphism one easily checks that f(0) = 0 and f(1) = 1. Then for each n ∈ IN* we deduce f(n) = f(1 + .. + 1) = f(1) + .. + f(1) = nf(1) = n. Moreover, f(-­n) = -f(n) = -n so that f(m) = m. ∀m ∈ ℤ. Further, for each n ∈ ℤ*, $$ 1 = f(1) = f(n \cdot \frac{1}{n}) = f(n) \cdot f(\frac{1}{n}) = nf(\frac{1}{n}) $$ and hence $$f(\frac{1}{n}) = \frac{1}{n}$$. Finally, for each$$\frac{m}{n} \in $$ we haver $$f(\frac{m}{n}) = f(m \cdot \frac{1}{n}) = f(m) \cdot f(\frac{1}{n}) = m \cdot \frac{1}{n} = \frac{m}{n}$$.

Indeed, the map f : $$ \mathbb{Z}{{ \otimes }_{\mathbb{Z}}}{{\mathbb{Z}}_{n}} \to \mathbb{Z} \cdot {{\mathbb{Z}}_{n}},f(\sum\limits_{{i = 1}}^{n} {{{x}_{i}} \otimes {{{\bar{y}}}_{i}}} ) = \sum\limits_{{i = 1}}^{n} {{{x}_{i}}{{{\bar{y}}}_{i}}} $$ is readily seen to be a ℤ-isomorphism

ℤ[i] is noetherian together with ℤ. It is not artinian: $$ 3\mathbb{Z}\left[ i \right] \supset {{3}^{2}}\mathbb{Z}\left[ i \right] \supset .. \supset {{3}^{n}}\mathbb{Z}\left[ i \right] \supset .. $$ is an infinite decreasing sequence of ideals in ℤ[i]

If R is a division ring (this is also true for semisimple rings, see next chapter) surely s(R) = R ≠ 0

If K is any field, an infinite product KI is a nonsemisimple ring. But it is the product of the (nonisomorphic) simple right ideals $${\left\{ {{P_i}} \right\}_{i \in I}}$$ where P _i and q _i : K → KI are the canonical injections. Indeed, it suffices to remark that isomorphic rings (or even modules) have the same annihilator ideal.

We have already used N(xy) = N(x).N(y) in the previous chapter, for the “norm” N(a + bi) = a2 + b2 in ℤ[i]. The units in ℤ[i] being {±1, ±I} i.e. the elements x ∈ ℤ[i] with N(x) = 1 we infer that x is prime in ℤ[i] iff N(x) is prime in ℤ (indeed, the no-associated decompositions correspond to each other). Hence N(1 + i) = 2 implies that the ideal (1 + i) is prime.

The polynomial f = X2 - 1̄ has 4 zeros in ℤ₁₅: 1̄, 4̄, 11̄, 14̄.

ℤ being an integral domain, for each multiplicative system S ⊂ ℤ, the ring of quotients identified with a subring with identity of Q, the rational numbers of the form $$ \frac{m}{s} $$ with m ∈ ℤ and s ∈ S. Now, if A is a subring of Q and 1 ∈ A let $$ {{S}_{A}} = \left\{ {n \in \mathbb{Z}\left| {\frac{1}{n} \in A} \right.} \right\} $$. Obviously, S_A is a multiplicative system and $$ {{\mathbb{Z}}_{{{{S}_{A}}}}} \subseteq \mathbb{Q} $$. Conversely, if $$ \frac{m}{n} \in A $$ with (m;n) = 1 let u, v ∈ such that um + vn = 1. Then $$ u\frac{m}{n} + v = \frac{1}{n} \in A $$ and hence A ∈ ℤ _SA .