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A simple example to show that this is not in general true without the orthogonality condition: Let $R$ be a ring of characteristic $2$ with two conjugate but non-equal idempotents $e$ and $f$, so that $Re$ and $Rf$ are isomorphic as left $R$-modules. For example, take $R=M_2(\mathbb{F}_2)$, $e=\begin{pmatrix}1&0\\0&0\end{pmatrix}$ and ...

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Motivation for the Jacobson radical Well, it sort of proves its own usefulness by being at the heart of so many algebraic theorems. But if you insist, there are a few good reasons that it is interesting. For one thing, it is the largest ideal such that R/J has “the same simple right(/left) modules.” Looked at another way, it is the set of elements that ...

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If there is a nontrivial central idempotent $e$, then $R=eRe\oplus (1-e)R(1-e)$ is a decomposition into two rings. If $M$ is a maximal left ideal in $eRe$ and $N$ a maximal left ideal of $(1-e)R(1-e)$, then $I=M\oplus (1-e)R(1-e)$ and $J=eRe\oplus N$ are maximal left ideals of $R$. Furthermore, $R/I$ and $R/J$ are nonisomorphic simple modules. (For ...

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In fact, much more is true: If $f:R\to S$ is a ring homomorphism, and $S$ has IBN property, then $R$ also has the IBN property. This follows immediately form the following characterization of IBN rings: a ring $R$ has the IBN property iff for $A\in M_{m\times n}(R)$ and $B\in M_{n\times m}(R)$ with the property $AB=I_m$ and $BA=I_n$ it follows $m=n$. ...

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If you know that $e_ie_j=0$ for $i\ne j$ and $e′_ie′_j=0$ for $i\ne j$ (a condition forgotten to mention by the OP), then $a=\sum_i \phi_i^{-1}(e′_i)$ and $a^{−1}=\sum_i \phi_i(e_i)$ will do the job, where $\phi_i:R e_i\to Re′_i$ are the given left $R$-module isomorphisms.

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As it is already commented by Mathematician 42 and san, this statement is usually stated with an orthogonality hypothesis (you can find, for example, Exercise 7 in here). So it is likely that your professor forget to mention about it. You (and perhaps your professor) should be careful that the definition of an idempotent decomposition $e = e_1 + \dotsb + ... 1 There are several possible definitions. One of them is to choose an embedding$K\to \overline{K}$into an algebraic closure : then$A\otimes_K \overline{K}$is isomorphic to$M_r(\overline{K})$and$\operatorname{End}_A(P)\otimes_K \overline{K}\simeq M_n(\overline{K})$. Then the reduced norm is the composition of$\operatorname{End}_A(P)\to ...

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McCoy's The Theory of Rings. Herstein's Noncommutative Rings. Warner's Modern Algebra. Bresar's Introduction to Noncommutative Algebra.

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