David Cameron has insisted that Britain would not give up on bringing Alexander Litvinenko's killer to justice as he kicked off his visit to Russia.
But the Prime Minister said the two governments had to end the "tit-for-tat culture" and work together despite festering tensions over the dissident's murder five years ago.
Mr Cameron is the first UK leader to visit Moscow since Tony Blair in 2005.
Relations nosedived following that trip when Mr Litvinenko was poisoned in London and Russia refused to extradite the prime suspect, ex-KGB agent Andrei Lugovoy.
In a speech at Moscow State University, Mr Cameron directly referred to the case and admitted there were "difficult issues that hamper mutual trust and co-operation".
He said: "We still disagree with you over the Litvinenko case. Our approach is simple and principled - when a crime is committed, that is a matter for the courts. It is their job to examine the evidence impartially and to determine innocence or guilt."
Mr Cameron said victims and their families had a "right to justice" and went on: "It is the job of governments to help courts to do their work, and that will continue to be our approach. So, we can't pretend these differences don't exist.
"We need to keep working for an honest and open dialogue to address them candidly. But, at the same time, we have a responsibility to recognise the many ways in which we do need each other, to end the old culture of tit for tat and find ways for us to work together to advance our mutual interests."
Mr Cameron's one-day visit is focusing on strengthening business links and improving conditions for UK firms in the country. BP chairman Bob Dudley is among 24 senior executives travelling with the premier, and some £215 million of deals are due to be sealed, creating 500 UK jobs.
As well as talks with President Dmitry Medvedev, Mr Cameron is set to meet Vladimir Putin - who could take over the Kremlin again in elections next year.