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Oh, goodness no. This isn't true over any base ring, meaning it's false even for vector spaces and even for plain old abelian groups.To be clear, since the wording is slightly confusing: if [math]\varphi:M\to M[/math] is a surjective endomorphism, then the induced homomorphism [math]\bar{\varphi}:M/\ker\varphi\to M[/math] is certainly an isomorphism. But it's definitely not true that the kernel must be trivial, so [math]\varphi[/math] itself need not be an isomorphism.In fact, take any module [math]N[/math] over any ring whatsoever, and form the direct product[math]M=N\oplus N\oplus N\oplus N\oplus \ldots[/math](BTW you can use either a direct sum or direct product here, doesn't matter). Let [math]\varphi:M\to M[/math] be the “left shift” operator[math]\varphi(n_1,n_2,n_3,\ldots)=(n_2,n_3,\ldots)[/math]Do you see what this does? Everyone steps over to the left and [math]n_1[/math] goes bye bye.Is this surjective? What is the kernel?As simple special cases you can use the left-shift operator on [math]\mathbb{R}^\mathbb{N}[/math], a real vector space, or [math]\mathbb{Z}^\mathbb{N}[/math], an abelian group.

How does an "operating system kernel" differ from "an operating system"?Informally, you can think of an operating system as divided into the kernel and the runtime system. In most systems, the kernel is always in memory and always in a position to run. Almost all of it executes in a privileged mode that gives it special access to the computer hardware to do things like manage user access to hardware (files on disk, I/O devices like a display, the network, keyboards, displays, memory and even CPUs). Typically, the kernel has a fairly limited API focused on managing the hardware and processes. The kernel's API describes the basic functionality that defines the operating system.The runtime system is the "face" the operating system gives to the user and most programmers: a lot of the user interface, such as command-lines, windows, menus, scroll-bars; convenience functions for programmers like buffered I/O, formatting functions for output and input, object management systems, graphics formatting systems. Much of it runs in user-mode. It is possible for an operating system kernel to have many runtime system "personalities". As the runtime system is what we see when we look at an operating system, it is often mistaken for the whole operating system, even though it is just a layer on top of the operating system kernel.

Step 1: Power On and System StartupWhen we press the power button, the Boot ROM code starts executing from a pre-defined location which is hardwired in ROM. It loads the Bootloader into RAM and starts executing.Step 2: BootloaderThe bootloader is a small program which runs before Android does. This is NOT part of the Android operating system. The bootloader is the place where manufacturer puts their locks and restrictions.The bootloader executes in two stages. In the first stage it detects external RAM and loads a program which helps in the second stage.In the second stage, the bootloader setups the network, memory, etc, which requires to run kernel. The bootloader is able to provide configuration parameters or inputs to the kernel for specific purposes.The bootloader can be found at:<android source>/bootable/bootloader/legacy/usbloaderThis legacy loader contains 2 important files:1- Init.s :: Initializes stacks, zeros the BSS segments and call_main() in main.c2- Main.c :: Initializes hardware (clocks, board, keyboard, console) and creates linux tags.Step 3: KernelThe Android kernel starts in a similar way as the linux kernel. As the kernel launches, is starts to setup cache, protected memory, scheduling and loads drivers. When the kernel finishes the system setup, it looks for “init” in the system files.What is the difference between the linux and android kernels?, here's a list of changes/addons that the Android Project made to the Linux kernel:Binder: It is an Android specific interprocess communication mechanism and remote method invocation system.ashmem: "Android Shared Memory". It is a new shared memory allocator, similar to POSIX SHM but with a different behavior and sporting a simpler file-based API.pmem: "Process memory allocator": It is used to manage large (1-16+ MB) physically contigous regions of memory shared between userspace and kernel drivers.logger: This is the kernel support for the logcat command.wakelocks: It is used for power management files. It holds the machine awake on a per-event basis until wakelock is released.oom handling: It kills processes as available memory becomes low.alarm manager: It lets user space tell the kernel when it would like to wake up.RAM_CONSOLE: Allows to save kernel printk messages to a buffer in RAM, so that after a kernel panic they can be viewed in the next kernel invocation.USB gadget driver for ADByaffs2 flash filesystemStep 4: init processInit is the very first process, we can say it is a root process, or the grandfather of all processes. The init process has two responsibilities.1- Mounts directories like /sys , /dev or /proc2- Runs init.rc script- The init process can be found at /init :: <android source>/system/core/init- Init.rc file can be found at :: <android source>/system/core/rootdir/Android has specific format and rules for init.rc files. More information about this rules can be found in: What is inside the init.rc and what is it used for.At this stage, you can finally see the Android logo in your screen.Step 5: Zygote and DalvikIn Java, we know that a separate Virtual Machine instance will popup in memory for separate per app, but in the case of Android, the VM should run as quick as possible for an app. But what happens if you have several apps thus launching several instances of the Dalvik (VM)?, it would consume an immense amount of memory.To overcome this problem, the Android OS has a system called “Zygote”. The Zygote enables code sharing across the Dalvik VM, achieving a lower memory footprint and minimal startup time. Zygote is a virtual machine process that starts at system boot. The Zygote preloads and initializes core library classes.The Zygote loading process:Load Zygote Init class:<android source>/frameworks/base/core/java/com/android/internal/os/ZygoteInit.javaregisterZygoteSocket() :: It registers a server socket for zygote command connections.preloadClasses() :: Is a simple text file that contains a list of classes that need to be preloaded, you can find the file at <android source>/framework/basepreloadResources() :: Everything that is included in the android.R file will be loaded with this method (themes and layouts).At this time, you can see the boot animation.Step 6: System serviceAfter the above steps are completed, Zygote launches the system services. The Zygote forks a new process to launch the system services.Core services:Starting power managerCreating the Activity ManagerStarting telephony registryStarting package managerSet activity manager service as system processStarting context managerStarting system contact providersStarting battery serviceStarting alarm managerStarting sensor serviceStarting window managerStarting Bluetooth serviceStarting mount serviceOther services:Starting status bar serviceStarting hardware serviceStarting NetStat serviceStarting connectivity serviceStarting Notification ManagerStarting DeviceStorageMonitor serviceStarting Location ManagerStarting Search ServiceStarting Clipboard ServiceStarting checkin serviceStarting Wallpaper serviceStarting Audio ServiceStarting HeadsetObserverStarting AdbSettingsObserverNow we have finally completed the booting process (system service are up and running in memory).Need to analyze the Android Bootup?The logcat :: Use adb to get the booting process events from the logcat.‘adb logcat –d –b events | grep “boot”‘adb logcat –d | grep preload’