Northern Ireland extraditions aim to tighten net around guerrillas. ANGLO-IRISH COOPERATION

Two convicted criminals, both anti-British activists, were extradited to Northern Ireland last week in an unusual example of cooperation between judicial authorities on either side of the Irish border. On Saturday, when the latest extradition occurred, street violence inspired by the outlawed Irish Republican Army (IRA) erupted in protest in the predominantly Roman Catholic neighborhoods of West Belfast and in parts of Derry, Northern Ireland's second-largest city.

Dozens of bombings and shooting incidents took place, more than 50 cars were hijacked, and many vehicles were burned on the streets. Local residents insisted that a only small number of people were responsible, but it was the worst street violence in several years.

The extraditions are part of what security officials hope is a tightening of the net around Irish guerrillas who have waged continuous war against British rule in the six counties of Northern Ireland since 1969. At least six more extraditions are pending.

Last week's extraditions were the first of anti-British activists since 1985. Even though they involved minor figures already convicted of crimes, there was strong opposition from supporters of Sinn Fein, the IRA's political wing.

Several hundred Sinn Fein supporters skirmished with police at the border crossing point Saturday when the Irish Garda Siochana handed over Robert Russell to the Royal Ulster Constabulary. Mr. Russell was convicted of attempted murder of an RUC superintendent in 1979. He had served only part of his 20-year sentence in Belfast's Maze prison when he escaped in 1983 and sought refuge in the Republic of Ireland.

Shouting ``Charlie Haughey is a traitor,'' Russell was quickly moved into a police helicopter and returned to the Maze, eluding Sinn Fein protesters who tried to block the road at the border. Charles Haughey, the Irish prime minister, is under pressure from IRA sympathizers opposed to cooperation with the British authorities.

Despite past sympathy with the IRA cause, Mr. Haughey has supported the Anglo-Irish agreement and cooperated with British security forces in policing border areas and prosecuting IRA members. The IRA is outlawed in the Republic of Ireland as well as in Northern Ireland.

Some observers say last week's extraditions are not true tests of the Irish government's willingness to cooperate in controlling violence in Northern Ireland since the the cases against both men were so clear-cut. ``It's difficult to see how if Russell hadn't been extradited, anyone could conceivably have been extradited,'' Robert McCartney, a Belfast lawyer, said in a British Broadcasting Corporation radio interview.

Mr. McCartney pointed to the long history of attempts to extradite criminal suspects, which have been frustrated by misunderstanding and lack of cooperation on both sides.

Britain has requested the return of several dozen suspects in connection with terrorist offenses in the past four years. Most have been denied on technical grounds by Ireland's highly independent judiciary, which has routinely set aside British warrants. A new extradition agreement was signed last year which broadens the grounds for extradition, but it also gives Irish judges greater discretion in granting the requests.

Earlier in the week, the IRA exploded a 300-pound car bomb on a narrow street in downtown Belfast, causing an estimated $7 million in property damage to stores and office buildings. An advance warning prevented injuries, but the explosion brought strong criticism from many local Catholics as well as Protestants who have enjoyed the relaxed atmosphere and shopping life of the city center.

Despite the incident, city merchants celebrated last Thursday the sixth anniversary of late-night shopping once a week. It was a happy sign of normality in a part of the city that in the early 1970s was a deserted zone of burned out storefronts and police checkpoints at every corner.

``If the IRA starts bombing downtown again, they'll lose out among the Catholics,'' a Catholic businessman from West Belfast said. Other Belfast observers agreed that it would be counterproductive to destroy jobs and disrupt the commercial life of the city, especially offices and businesses where many Catholics are employed.

Gerry Adams, leader of Sinn Fein, told a press conference last week that the IRA campaign would continue until British troops leave Northern Ireland. Coming after the killing of eight British soldiers in a land-mine explosion on Aug. 20, and the murder of a Royal Navy recruiting officer on Aug. 22, his remarks were interpreted as a prediction of more violence.

But government officials in Belfast said that it was hypocritical of Mr. Adams to complain about the high unemployment rate among Catholics if the IRA was going to attack the city's commercial life and plunge Belfast back into the dismal years of the 1970s.