i have configured in my lab the following scenario:
R1, R2 and R3 are all configured with RIPv2 and with the commands version 2 and no auto-summary, the network statement are as follow:
R1: network 10.1.12.0
R2: network 10.1.12.0 and 10.1.23.0
R3: network 10.1.23.0
Despite this configuration R3 is advertising also network 10.1.34.0 and under RIP database i can see these addresses with auto-summary.
what is the reason for this behavior and why R3 is advertising a network which i don't want?
The behavior you have observed is correct. Even though you configured each router with a particular network statement, the RIP configuration now contains only the network 10.0.0.0 statement - check your running-config. Cisco's RIP implementation always masks the address in the network command by the corresponding classful mask before entering it into running configuration. The roots of this behavior lie in RIPv1 and the behavior was not changed for RIPv2. This means that even if you configure each RIPv2 router using a particular subnet of the 10.0.0.0/8 range, the RIP on each router is running over all interfaces that lie in the 10.0.0.0/8 range. This is the reason why R3 advertises the 10.1.34.0/x network even though you have not explicitly added it to RIP. There is nothing you can do about it - in other words, Cisco's RIP implementation can not be configured to run only over a selected subnet of a classful network. The automatic summarization is not related to this behavior at all.
The auto-summary networks in the RIP database are always created by the RIP process. However, they are not advertised until the condition for their advertisement is met: a network from one particular classful network is going to be advertised out an interface that itself lies in a different classful network.
Please feel welcome to ask further.