2.5. Quotients 15 2.4. Ideals A left ideal of an algebra A is a subspace I ⊆ A such that aI ⊆ I for all a ∈ A. Similarly, a right ideal of an algebra A is a subspace I ⊆ A such that Ia ⊆ I for all a ∈ A. A two-sided ideal is a subspace that is both a left and a right ideal. Left ideals are the same as subrepresentations of the regular rep- resentation A. Right ideals are the same as subrepresentations of the regular representation of the opposite algebra Aop. Below are some examples of ideals: • If A is any algebra, 0 and A are two-sided ideals. An algebra A is called simple if 0 and A are its only two-sided ideals. • If φ : A → B is a homomorphism of algebras, then ker φ is a two-sided ideal of A. • If S is any subset of an algebra A, then the two-sided ideal generated by S is denoted by S and is the span of ele- ments of the form asb, where a, b ∈ A and s ∈ S. Similarly, we can define S = span{as} and S r = span{sb}, the left, respectively right, ideal generated by S. Problem 2.4.1. A maximal ideal in a ring A is an ideal I = A such that any strictly larger ideal coincides with A. (This definition is made for left, right, or two-sided ideals.) Show that any unital ring has a maximal left, right, and two-sided ideal. (Hint: Use Zorn’s lemma.) 2.5. Quotients Let A be an algebra and let I be a two-sided ideal in A. Then A/I is the set of (additive) cosets of I. Let π : A → A/I be the quotient map. We can define multiplication in A/I by π(a) · π(b) := π(ab). This is well defined because if π(a) = π(a ), then π(a b) = π(ab + (a − a)b) = π(ab) + π((a − a)b) = π(ab) because (a − a)b ∈ Ib ⊆ I = ker π, as I is a right ideal similarly, if π(b) = π(b ), then π(ab ) = π(ab + a(b − b)) = π(ab) + π(a(b − b)) = π(ab)

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